chunk_uid stringlengths 40 40 | chunk_type stringclasses 2
values | chunk_index int64 0 6.71k | total_chunks int64 1 6.71k | section_title stringlengths 1 157 | embed_text stringlengths 1 83.3k | spans dict | paper_doi stringlengths 0 63 | paper_id_arxiv stringlengths 9 16 | title stringlengths 7 245 | authors listlengths 1 768 | categories listlengths 1 7 | year int64 2k 2.02k | language stringclasses 2
values | discipline stringclasses 8
values | dense_vector listlengths 1.02k 1.02k |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
a6337b895c85dea95f84bba26f6f062e899eb23a | subsection | 120 | 130 | M-theory parameterization | If we consider the E_{8(8)} EFT, the generalized metric additionally contains the dual graviton A_{i_1\cdots i_8,\,j} although the explicit parameterization in our convention is not determined yet.We can also parameterize the same \mathcal {M}_{IJ} in terms of the dual fields, \tilde{G}_{ij}, \Omega ^{i_1i_2i_3}, and \... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 198,
"openalex_id": "",
"raw": "H. Godazgar, M. Godazgar and M. J. Perry, “E_8 duality and dual gravity,” JHEP 1306, 044 (2013) [arXiv:1303.2035 [hep-th]].",
"source_ref_id": "b9258de251e0a9456446381d63defcfde9caa4a0",
... | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.04311491176486015,
0.019146926701068878,
-0.017133066430687904,
-0.015279400162398815,
-0.004329036455601454,
-0.028041476383805275,
-0.011205910705029964,
0.014753051102161407,
0.027385445311665535,
0.012006877921521664,
-0.053886014968156815,
0.027095571160316467,
0.0014894173946231604,... |
d12a0ba01e9a9adc9f2a42d1cc3e263d014b3583 | subsection | 121 | 130 | Type IIB parameterization | When we consider type IIB theory, we parameterize the generalized metric as , (see also for the case of \mathrm {SL}(5) EFT)\begin{split}
&\mathcal {M}_{IJ} = (\mathsf {L}_6^\mathrm {T}\,\mathsf {L}_4^\mathrm {T}\,\mathsf {L}_2^\mathrm {T}\,\hat{\mathsf {M}}\,\mathsf {L}_2\,\mathsf {L}_4\,\mathsf {L}_6)_{IJ} \,, \qquad... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/jhep10(2014)009",
"end": 5582,
"openalex_id": "https://openalex.org/W3099268559",
"raw": "A. G. Tumanov and P. West, “Generalised vielbeins and non-linear realisations,” JHEP 1410, 009 (2014) [arXiv:1405.7894 [hep-th]].",
"sou... | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.05302678048610687,
0.013287205249071121,
-0.03246288746595383,
-0.01749761775135994,
0.01241766382008791,
-0.0060105156153440475,
-0.017146749421954155,
0.019938435405492783,
0.011868479661643505,
0.02080797776579857,
-0.04597891867160797,
0.004443052224814892,
-0.0034419354051351547,
0... |
6f8f5ea8b5a6d48eaf12a9b8bdd5ad332bb622af | subsection | 122 | 130 | Type IIB parameterization | The metric g^{\text{\tiny E}}_{mn} is the standard Einstein-frame metric and \Phi is the standard dilaton and the string-frame metric is defined as g_{mn} \equiv \operatorname{e}^{\frac{1}{2}\,\Phi } g^{\text{\tiny E}}_{mn} .
Other fields are further parameterized as\begin{split}
&\bigl (m_{\alpha \beta }\bigr ) \equiv... | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.06745922565460205,
-0.0164679866284132,
-0.010744636878371239,
-0.038308292627334595,
0.018940474838018417,
-0.053845278918743134,
0.0036915610544383526,
0.030417699366807938,
0.02699895203113556,
0.020161455497145653,
-0.05720297992229462,
0.03623262420296669,
-0.01573539897799492,
0.0... |
f0f635b2d06ea3878bf1684e95892591e98afdca | subsection | 123 | 130 | Type IIB parameterization | \end{split}Then, \Phi , B_{mn}, C_{m_1\cdots m_{2n}}, and D_{m_1\cdots m_6} are the standard dilaton, the B-field, the R–R potentials, and the dual potential of the B-field.We can also provide the non-geometric parameterization as \begin{split}
&\mathcal {M}_{IJ} = (\tilde{\mathsf {L}}_6^\mathrm {T}\,\tilde{\mathsf {L}... | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.041697196662425995,
0.028693532571196556,
0.009752748534083366,
-0.0023408886045217514,
-0.003521825885400176,
-0.015491454862058163,
0.029029307886958122,
0.03382173925638199,
0.03351648896932602,
0.010966118425130844,
-0.06398047506809235,
0.04987790435552597,
-0.0013507327530533075,
... |
5377ad4bf6a03ac57b2e44e89bf87001bc5fec04 | subsection | 124 | 130 | Type IIB parameterization | \end{split}By using the dual metric g^{\text{\tiny E}}_{mn} and the dual dilaton \Phi , we define the dual string-frame metric as \tilde{g}_{mn} \equiv \operatorname{e}^{\frac{1}{2}\,\tilde{\phi }} \tilde{g}^{\text{\tiny E}}_{mn} .
Again, we further parameterize other fields as\begin{split}
&\bigl (\tilde{m}_{\alpha \... | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.0669078454375267,
-0.015437336638569832,
0.0003708146687131375,
-0.01788686029613018,
-0.008325326256453991,
-0.04081012308597565,
0.02947058528661728,
0.00417410908266902,
0.03131726756691933,
0.040352266281843185,
-0.04932621866464615,
0.03522429242730141,
-0.020038776099681854,
0.009... |
5878331361da716e3387573bbd2168c4803a2bfa | subsection | 125 | 130 | Contents of the | In this appendix, we provide a list of branes contained in the p-brane multiplets. | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
0.01681164652109146,
0.030923055484890938,
-0.016643835231661797,
0.023951256647706032,
-0.033409714698791504,
-0.007204446475952864,
0.06608716398477554,
-0.0040503558702766895,
0.04576673358678818,
-0.02151036448776722,
-0.024714035913348198,
0.023280011489987373,
-0.0390847884118557,
0.... |
c006468708483b5475ac22fded46a8cb9a932e5e | subsection | 126 | 130 | “Elementary” branes | We first provide a list of “elementary” branes that are connected to the standard branes.
[Table: Branes in the particle multiplet.][Table: Branes in the string multiplet.][Table: Branes in the membrane multiplet.][Table: Branes in the 3-brane multiplet.][Table: Branes in the 4-brane multiplet.][Table: Branes in the 5-... | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
0.01012435369193554,
0.017654675990343094,
-0.03482106328010559,
-0.011238261125981808,
-0.024063460528850555,
-0.060700345784425735,
0.03460743650794029,
-0.040741559118032455,
0.019211094826459885,
-0.008010980673134327,
-0.019089022651314735,
0.008270383812487125,
-0.020920103415846825,
... |
430cb2184c17b745d26c4223cb4d4a9578dfeac4 | subsection | 127 | 130 | Missing states | We here provide a list of the missing states in each multiplet.
[Table: Missing states in the string multiplet.][Table: Missing states in the membrane multiplet.][Table: Missing states in the 3-brane multiplet.][Table: Missing states in the 4-brane multiplet.][Table: Missing states in the 5-brane multiplet.][Table: Mis... | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.0033109067007899284,
-0.007827166467905045,
-0.03704553470015526,
-0.012206107378005981,
-0.0011166681069880724,
-0.020353684201836586,
0.03939520940184593,
-0.00035903119714930654,
-0.028577549383044243,
0.008529017679393291,
0.0025346744805574417,
-0.016417214646935463,
-0.0163866989314... |
423329fe24cd65e66c96cd1dcfac4fd2f7a1722e | subsection | 128 | 130 | Counting of mixed-symmetry potentials | In a series of work on the mixed-symmetry potentials , , , , the number of supersymmetric branes that couple to the mixed-symmetry potentials has been counted up to \alpha =-7 .
We reproduce the same results by counting the number of branes contained in Tables REF –REF .
Our results include all of the “elementary” exot... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1016/j.physletb.2011.07.009",
"end": 177,
"openalex_id": "https://openalex.org/W2035128568",
"raw": "E. A. Bergshoeff and F. Riccioni, “Dual doubled geometry,” Phys. Lett. B 702, 281 (2011) [arXiv:1106.0212 [hep-th]].",
"source_ref... | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
0.018672281876206398,
0.00903866533190012,
-0.008680170401930809,
-0.014606788754463196,
-0.024514997377991676,
0.0014864219119772315,
0.06150869280099869,
-0.015064442530274391,
0.021845348179340363,
-0.00048435042845085263,
-0.05781694874167442,
0.038442932069301605,
-0.0434466153383255,
... |
52105f5b45bb58614e1b4d3999020923249d581f | subsection | 129 | 130 | Counting of mixed-symmetry potentials | Repeating a similar argument, we obtain the set of Tables REF –REF .
[Table: Number of F-branes in the p-brane multiplet in d-dimensions .][Table: Number of D-branes in the p-brane multiplet in d-dimensions .][Table: Number of S-branes in the p-brane multiplet in d-dimensions .][Table: Number of E-branes in the p-brane... | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
0.026975108310580254,
0.014738662168383598,
-0.04967813938856125,
-0.014326712116599083,
-0.012869629077613354,
0.014944637194275856,
0.04250715672969818,
-0.017256135120987892,
0.013159519992768764,
0.013357865624129772,
-0.01907176710665226,
0.01343415305018425,
-0.0377163290977478,
0.03... |
c109c0fb17c0ac5ac68ba7aa8d141074d8f4343c | abstract | 0 | 71 | Abstract | Modeling flow through porous media with multiple pore-networks has now become
an active area of research due to recent technological endeavors like
geological carbon sequestration and recovery of hydrocarbons from tight rock
formations. Herein, we consider the double porosity/permeability (DPP) model,
which describes t... | {
"cite_spans": []
} | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.018370134755969048,
-0.02886735461652279,
-0.037625208497047424,
-0.0003540236793924123,
0.010413303039968014,
0.015349132008850574,
0.0029542502015829086,
0.018202301114797592,
-0.0015591346891596913,
0.04833603650331497,
-0.03201041743159294,
-0.004489544779062271,
0.03280381113290787,
... |
b280c98e7479d6e6785e2566c786528d45f7c6da | subsection | 1 | 71 | A list of abbreviations and symbols | [c]|p.2 || p.75|
2| c |AbbreviationsCG Continuous GalerkinDG Discontinuous GalerkinDPP Double porosity/permeability2|c|Symbols in the DPP model,
\mathsection \ref {Sec:S2_DG_GE}\Omega , \overline{\Omega }, \partial \Omega
Computational porous domain, its set closure,
and its boundary\mathbf {u}_1, \mathbf {u}_2, p_1,... | {
"cite_spans": []
} | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.04046599194407463,
-0.01864609308540821,
-0.044372208416461945,
-0.02423076704144478,
-0.0029163130093365908,
-0.04342617094516754,
-0.019119111821055412,
0.016738759353756905,
-0.015472290106117725,
0.025802409276366234,
-0.02735879272222519,
-0.0007610261091031134,
0.022994814440608025,... |
d1e5c48a78f2b56a372238ac082fbbebd34ec483 | subsection | 2 | 71 | A list of abbreviations and symbols | (REF )\mathcal {T}, \mathcal {T}_{h} A mesh, and a mesh
with mesh-size h2|c|Symbols in the proposed
DG formulation,
\mathsection \ref {Sec:S3_DG_Mixed}(\cdot ;\cdot )_{\mathcal {K}}, (\cdot ;\cdot )
L_2 inner-products over \mathcal {K}
and \widetilde{\Omega }, respectively\Vert \cdot \Vert _{\mathcal {K}}, \Vert \cdot... | {
"cite_spans": []
} | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.03887559846043587,
0.014616493135690689,
-0.03133849427103996,
-0.015776047483086586,
0.009390868246555328,
0.0011013862676918507,
0.0017545894952490926,
0.02459476701915264,
-0.005500256549566984,
0.021787423640489578,
-0.011999865993857384,
0.02056683972477913,
0.01264830119907856,
0.... |
a588b641fcae08da28c1bc1ed13ac25670073843 | subsection | 3 | 71 | A list of abbreviations and symbols | (REF )
& ()2|c|Other symbols{P}^{m}(\omega ) Set of all
polynomials over \omega up to and
including m-th order,
\mathsection \ref {Subsec:DG_functional_analysis}c, D Concentration and diffusivity,
\mathsection \ref {Sec:S8_DG_NR}m(\omega ) Net rate of volumetric flux from element \omega ,
\mathsection \ref {Sec:Element... | {
"cite_spans": []
} | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
0.0005651991232298315,
-0.018559519201517105,
-0.06483621150255203,
-0.012271260842680931,
-0.017628489062190056,
-0.06813296675682068,
-0.004964213352650404,
0.024054111912846565,
0.01227889209985733,
-0.001497658551670611,
-0.008898190222680569,
0.0054907784797251225,
-0.009569752030074596... |
7758bf06dfbafe2e78a03356ec20c8cd986e98fc | subsection | 4 | 71 | INTRODUCTION AND MOTIVATION | This paper presents a discontinuous Galerkin
version of the continuous stabilized mixed
formulation proposed recently by
for the double porosity/permeability (DPP)
mathematical model. The DPP model describes
the flow of a single-phase incompressible
fluid in a rigid porous medium with two
distinct pore-networks with p... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1016/j.cma.2018.04.004",
"end": 184,
"openalex_id": "https://openalex.org/W2618072050",
"raw": "S. H. S. Joodat, K. B. Nakshatrala, and R. Ballarini. Modeling flow in porous media with double porosity/permeability: A stabilized mixed for... | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.020600654184818268,
-0.000434425804996863,
-0.05063182860612869,
0.018845783546566963,
0.010712339542806149,
-0.010147728957235813,
-0.004303247667849064,
0.004734335467219353,
-0.012284093536436558,
0.053225982934236526,
-0.038576629012823105,
0.00851493701338768,
0.025834744796156883,
... |
60fe12593a93191a0bb7da7b6d9ddf999cb1e6dc | subsection | 5 | 71 | INTRODUCTION AND MOTIVATION | Although there is an on-going debate on using
H(div) elements vs. DG methods, the later do
enjoy some unique desirable features.
DG methods combine the attractive features of
both finite element and finite volume methods.
Application of completely discontinuous basis
functions in the form of piecewise polynomials
in DG... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1002/cnm.464",
"end": 560,
"openalex_id": "https://openalex.org/W2071505893",
"raw": "B. Rivière and M. F. Wheeler. Discontinuous Galerkin methods for flow and transport problems in porous media. International Journal for Numerical Metho... | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.005080919712781906,
-0.004432454239577055,
-0.054318543523550034,
0.02686936967074871,
0.013869537971913815,
0.0003197031619492918,
0.052975837141275406,
0.00317176035605371,
0.01488419622182846,
0.028379913419485092,
-0.055264540016651154,
0.022261448204517365,
0.003400630783289671,
0.... |
04718c70a8ce5230eaeb0842a9671732e88bd6b4 | subsection | 6 | 71 | INTRODUCTION AND MOTIVATION | In Section , the proposed DG
formulation is implemented to study viscous-fingering-type
physical instabilities in heterogeneous porous
media with double pore-networks.
Finally, conclusions are drawn in
Section .Throughout this paper, repeated indices do not imply summation. | {
"cite_spans": []
} | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.044982604682445526,
-0.015838515013456345,
-0.02825908549129963,
0.004550903104245663,
0.0019607411231845617,
-0.002603513188660145,
0.030334265902638435,
0.0347898043692112,
-0.0177916269749403,
0.02798442915081978,
-0.008705079555511475,
0.022949062287807465,
0.0359189435839653,
-0.01... |
d8144eb3a0736ff2554c4e524823c1e03c12e473 | subsection | 7 | 71 | Governing
equations | The DPP model deals with the flow of
a single-phase incompressible fluid
through a rigid porous medium with two
pore-networks exhibiting different
hydromechanical properties.
We refer to these two pore-networks as
macro-pore and micro-pore networks,
which are denoted by subscripts 1
and 2, respectively.
We denote the p... | {
"cite_spans": []
} | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.0306471586227417,
-0.0017571775242686272,
-0.04393420368432999,
-0.01359977200627327,
-0.009900450706481934,
0.022775612771511078,
-0.02338581159710884,
0.017024504020810127,
0.016536345705389977,
0.04762589931488037,
-0.03316422179341316,
0.01743638701736927,
0.030311549082398415,
-0.0... |
6c3f278552cd3d49ec47a7c03ca8e8b79d7ec08b | subsection | 8 | 71 | Governing
equations | We denote the viscosity and true density of the
fluid by \mu and \gamma , respectively.The abstract boundary value problem under
the DPP model takes the following form:
Find \mathbf {u}_{1}(\mathbf {x}),
\mathbf {u}_{2}(\mathbf {x}),
p_{1}(\mathbf {x}) and
p_{2}(\mathbf {x}) such that&\mu k_{1}^{-1} \mathbf {u}_1(\math... | {
"cite_spans": []
} | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.029809245839715004,
-0.03279932215809822,
-0.02819216251373291,
-0.034416407346725464,
-0.03322647884488106,
-0.001575130270794034,
-0.0014416446210816503,
0.027475154027342796,
0.023966388776898384,
0.04726153612136841,
-0.032951876521110535,
0.003613646375015378,
0.013226517476141453,
... |
56c85e65b6d87466f7fc93471fd13ca94ff20dd6 | subsection | 9 | 71 | Governing
equations | The
dimension of \chi (\mathbf {x})
is one over the time
[\mathrm {M}^{0}\mathrm {L}^{0}\mathrm {T}^{-1}].
\Gamma _{i}^{u} denotes that part
of the boundary on which the normal
component of velocity is prescribed
in the macro-pore (i=1) and micro-pore
(i=2) networks, and u_{n1}(\mathbf {x})
and u_{n2}(\mathbf {x}) deno... | {
"cite_spans": []
} | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.031079674139618874,
-0.009116399101912975,
-0.061823680996894836,
-0.030911840498447418,
-0.006114472169429064,
-0.020567206665873528,
0.026944871991872787,
0.05016689747571945,
0.002570900833234191,
0.003455840051174164,
-0.027723008766770363,
-0.002626209519803524,
-0.004447582177817821... |
b9001c8d6e76ac8a21c4ba11183c7bc693f921ff | subsection | 10 | 71 | Governing
equations | That is,0 <
\inf _{\mathbf {x} \in \Omega } \frac{\mu }{ k_i(\mathbf {x})}
\le \sup _{\mathbf {x} \in \Omega } \frac{\mu }{ k_i(\mathbf {x})}
< +\infty \qquad i = 1, 2This also means that there exist
two non-dimensional constants 1 \le \mathcal {C}_{\mathrm {drag},1}, \;
\mathcal {C}_{\mathrm {drag},2} < +\infty
where... | {
"cite_spans": []
} | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.07324918359518051,
0.005528024397790432,
-0.023256616666913033,
-0.010224365629255772,
-0.028765566647052765,
0.0032847682014107704,
-0.0012503864709287882,
0.017060955986380577,
0.03876102715730667,
-0.003050141967833042,
-0.0164810661226511,
-0.03219912201166153,
0.007828506641089916,
... |
cc22d2182929d828e132e2c88b2ba03d1c5cef38 | subsection | 11 | 71 | Geometrical definitions | The domain is partitioned into “Nele”
subdomains, which will be elements in the
context of the finite element method. These
elements form a mesh on the domain.
Mathematically, a mesh \mathcal {T}
on \Omega is a finite collection
of disjoint polyhedra \mathcal {T}
= \lbrace \omega ^{1}, \cdots , \omega ^{Nele}\rbrace
s... | {
"cite_spans": []
} | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.005958867724984884,
-0.004963180050253868,
-0.037447020411491394,
0.001760574639774859,
0.00555067416280508,
-0.048006653785705566,
0.020707257091999054,
0.020768295973539352,
0.002275585662573576,
0.03170941770076752,
-0.03283862769603729,
0.030946439132094383,
-0.0023423463571816683,
... |
515e82ad523772f44187c4c2dd1d1a250521790a | subsection | 12 | 71 | Average and jump operators | Consider an interior edge \Upsilon \in \mathcal {E}^{\mathrm {int}}. We denote the
elements that juxtapose \Upsilon by
\omega _{\Upsilon }^{+} and \omega _{\Upsilon }^{-}.
The unit normal vectors on this interior
edge pointing outwards to \omega ^{+}_{\Upsilon }
and \omega ^{-}_{\Upsilon } are, respectively,
denoted by... | {
"cite_spans": []
} | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.018246084451675415,
-0.004428031388670206,
-0.054616205394268036,
0.020717542618513107,
0.01701035350561142,
-0.03905516117811203,
0.0049810586497187614,
0.021861737594008446,
0.010473190806806087,
0.0019642000552266836,
-0.04796462133526802,
0.027918338775634766,
-0.018490178510546684,
... |
41869b9c3c3c8118e83bf9eaa984915da96e2a2a | subsection | 13 | 71 | Average and jump operators | Mathematically,\varphi ^{+}_{\Upsilon }(\mathbf {x}) :=
\varphi (\mathbf {x})\big |_{\partial \omega ^{+}_{\Upsilon }}
\quad \mathrm {and} \quad \varphi ^{-}_{\Upsilon }(\mathbf {x}) :=
\varphi (\mathbf {x}) \big |_{\partial \omega ^{-}_{\Upsilon }}
\quad \forall \mathbf {x} \in \UpsilonFor a vector field {\tau }
(\mat... | {
"cite_spans": []
} | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.00887431763112545,
0.008294396102428436,
-0.0516434907913208,
0.030598463490605354,
0.015871521085500717,
-0.009545804932713509,
0.027408897876739502,
0.03516152501106262,
0.028171950951218605,
0.0036760123912245035,
-0.027500463649630547,
0.02341049537062645,
-0.008065479807555676,
0.0... |
bd35c66847794661e63b46d718241a0b438c728c | subsection | 14 | 71 | Mesh-related quantities | We denote the element diameter
(i.e., the length of the largest edge)
of \omega \in \mathcal {T} by h_{\omega }.
The maximum element diameter in a given mesh
is referred to as the mesh-size and
is denoted by:h := \max _{\omega \in \mathcal {T}} h_{\omega }We denote the diameter of the inscribed
circle in \omega \in \ma... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1109/mcse.1999.10004",
"end": 1483,
"openalex_id": "https://openalex.org/W1524015806",
"raw": "D. Braess. Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics. Cambridge University Press, 2007.",
"source_ref_i... | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.0026895590126514435,
-0.006527426186949015,
-0.0396757610142231,
-0.0006447311607189476,
-0.005920844618231058,
-0.04187319055199623,
0.000686219020280987,
0.0019494533771649003,
0.016282321885228157,
0.014412984251976013,
-0.017701493576169014,
0.042727746069431305,
-0.01105580385774374,... |
aae1ff7bc572f26468d3842961f2eda6c44086de | subsection | 15 | 71 | Mesh-related quantities | The locally quasi-uniform condition
implies the following useful bound:
\frac{1}{2} \left(1 + \frac{1}{\mathcal {C}_{\mathrm {lqu}}}
\right) h_{\omega ^{+}_{\Upsilon }}
\le h_{\Upsilon } \le \frac{1}{2} \left(1 + \mathcal {C}_{\mathrm {lqu}}\right)
h_{\omega ^{+}_{\Upsilon }} \quad \forall \Upsilon \in \mathcal {E}^{\... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1016/j.cma.2005.06.018",
"end": 1312,
"openalex_id": "https://openalex.org/W1985247669",
"raw": "T. J. R. Hughes, A. Masud, and J. Wan. A stabilized mixed discontinuous Galerkin method for Darcy flow. Computer Methods in Applied Mechanic... | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.036772314459085464,
-0.007037854753434658,
-0.046598609536886215,
0.00537470867857337,
-0.00938380602747202,
-0.07354462891817093,
0.009246482513844967,
-0.015555757097899914,
-0.010825708508491516,
0.02079695649445057,
-0.04702583700418472,
0.051786404103040695,
0.029707757756114006,
0... |
1ce7e6b72bea9ca47f67abd07bf9ca4f1abcf76f | subsection | 16 | 71 | Functional analysis aspects | We introduce the following broken Sobolev
spaces (which are piece-wise discontinuous
spaces):\mathcal {U} &:= \left\lbrace \mathbf {u}(\mathbf {x}) \; \big | \;
\mathbf {u}(\mathbf {x})\big |_{\omega ^i} \in \left(L_{2}(\omega ^i)
\right)^{nd}; \; \mathrm {div}[\mathbf {u}] \in L_{2}(\omega ^i); \;
i = 1, \cdots , Nele... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/bfb0082920",
"end": 1460,
"openalex_id": "https://openalex.org/W3021722416",
"raw": "L. C. Evans. Partial Differential Equations. American Mathematical Society, Providence, Rhode Island, 1998.",
"source_ref_id": "104c775ce5382... | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.04570142924785614,
0.024406637996435165,
-0.03453539311885834,
-0.014796524308621883,
0.0461590513586998,
-0.0044542113319039345,
0.047501418739557266,
0.07218263298273087,
0.01656600460410118,
0.02062360942363739,
-0.0014548643957823515,
-0.011654169298708439,
0.011295696720480919,
-0.... |
7c410e640d8532365da3529b765bbdac51d313ee | subsection | 17 | 71 | Functional analysis aspects | However, this condition is
seldom employed in a numerical implementation.
Alternatively, one can prescribe the pressure
on a portion of the boundary in one of the
pore-networks. For further details refer
to .We denote the standard L_2 inner-product over
a set \mathcal {K} by (\cdot ;\cdot )_{\mathcal {K}}.
That is,(a;b... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1016/j.cma.2018.04.004",
"end": 208,
"openalex_id": "https://openalex.org/W2618072050",
"raw": "S. H. S. Joodat, K. B. Nakshatrala, and R. Ballarini. Modeling flow in porous media with double porosity/permeability: A stabilized mixed for... | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.0662771686911583,
0.010916509665548801,
-0.03231469914317131,
0.01904095523059368,
-0.0062897708266973495,
0.019849585369229317,
0.003970679827034473,
0.030850009992718697,
-0.0021741476375609636,
0.017072780057787895,
-0.025571025907993317,
0.003192564006894827,
0.01766780950129032,
0.... |
83476960a43d744d2c5cdb3d5a7fa1e09b6d1fdd | subsection | 18 | 71 | Functional analysis aspects | For a scalar function \varphi (\mathbf {x}) \in C^{\infty }_{c}(\mathcal {K}) (which is a set of
infinitely differentiable functions with compact
support in \mathcal {K}) , the
multi-index (classical) partial derivative with
respect to a given coordinate system \mathbf {x}
= (x_1, \cdots , x_{nd}) is defined as follows... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/bfb0082920",
"end": 1419,
"openalex_id": "https://openalex.org/W3021722416",
"raw": "L. C. Evans. Partial Differential Equations. American Mathematical Society, Providence, Rhode Island, 1998.",
"source_ref_id": "104c775ce5382... | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.04902613162994385,
-0.021246692165732384,
-0.059161048382520676,
-0.0002635326236486435,
-0.004689688328653574,
-0.002734061563387513,
0.03229741379618645,
0.03730381652712822,
0.04047860950231552,
0.0029496573843061924,
-0.008661995641887188,
0.01051650196313858,
-0.025444135069847107,
... |
39a8ee78f773d27b8b863bf4879017be3a48c237 | subsection | 19 | 71 | Inverse and trace
inequalities | The inequalities given below play a crucial role
in obtaining bounds on the error due to terms
defined on the element interface. Mathematical
proofs to these estimates can be found in
, , , .Lemma 2.1 (Continuous trace inequality.)
For an admissible mesh \mathcal {T}_{h}, the
following estimates hold \forall \omega \i... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1093/acprof:oso/9780199679423.001.0001",
"end": 191,
"openalex_id": "https://openalex.org/W253824085",
"raw": "R. Verfürth. A Posteriori Error Estimation Techniques for Finite Element Methods. Oxford Science Publications, New Jersey, 201... | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
0.014024169184267521,
-0.00550513481721282,
-0.02984904684126377,
0.005325826816260815,
0.00397529499605298,
-0.05243421718478203,
0.018068134784698486,
-0.018907448276877403,
0.022951414808630943,
0.016862574964761734,
-0.014840591698884964,
-0.009789450094103813,
0.009049328044056892,
0.... |
c9b070a2a8b824bdc7763f649dded76ec092849a | subsection | 20 | 71 | Inverse and trace
inequalities | Then
the following estimates hold \forall \omega \in \mathcal {T}_h:&\Vert \mathrm {grad}[v^{h}]\Vert _{\omega } \le \mathcal {C}_{\mathrm {inv}} h_{\omega }^{-1}
\Vert v^{h}\Vert _{\omega }
\quad \forall v^{h}(\mathbf {x}) \in H^{1}(\omega ) \cap {P}^{m}(\omega ) \\
&\Vert \mathrm {grad}[\mathbf {v}^{h}]\Vert _{\omega... | {
"cite_spans": []
} | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.012184850871562958,
0.006809629965573549,
-0.0332355722784996,
-0.012406115420162678,
0.017823301255702972,
-0.06500621885061264,
-0.000015498108041356318,
0.005367590580135584,
0.037355683743953705,
-0.008530158549547195,
-0.021470362320542336,
-0.003328516613692045,
-0.00178728939499706... |
5ba184264d513b473417d314e083eef10f1f4f0e | subsection | 21 | 71 | A STABILIZED MIXED DG FORMULATION | We propose a stabilized four-field formulation
for the DPP model. The proposed formulation
draws its inspiration from the stabilized two-field
formulations proposed by
, for Darcy equations,
which describe the flow of an incompressible
fluid through a porous medium with a single
pore-network. | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1016/j.cma.2005.06.018",
"end": 294,
"openalex_id": "https://openalex.org/W1985247669",
"raw": "T. J. R. Hughes, A. Masud, and J. Wan. A stabilized mixed discontinuous Galerkin method for Darcy flow. Computer Methods in Applied Mechanics... | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.008661293424665928,
0.032111648470163345,
-0.024740010499954224,
-0.0027662720531225204,
0.016986817121505737,
-0.003914752043783665,
0.01935245655477047,
0.0030524381436407566,
0.03043280728161335,
0.055706996470689774,
-0.030951721593737602,
-0.002848306205123663,
0.04694649949669838,
... |
152d33f2bd185ab4e5cb10ba4fd5042d06cf592a | subsection | 22 | 71 | Weak form in terms of numerical fluxes | Multiplying the governing equations
(REF )–()
by weighting functions, integrating over an element
\omega , and using equation () and
the divergence theorem, we
obtain the following:&\left( \mathbf {w}_1 ; \mu k_{1}^{-1} \mathbf {u}_{1} \right)_{\omega }
- \left(\mathrm {div}[\mathbf {w}_{1}] ; p_{1} \right)_{\omega }
+... | {
"cite_spans": []
} | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.01609804853796959,
0.019073517993092537,
-0.01568606123328209,
-0.04440315067768097,
-0.004665382672101259,
-0.025924725458025932,
0.035492002964019775,
0.051361169666051865,
-0.003173833480104804,
0.025268597528338432,
-0.021636998280882835,
0.007057201582938433,
-0.018737824633717537,
... |
b73306c5741014ce953e1c2b4264ea0b5d04de66 | subsection | 23 | 71 | Weak form in terms of numerical fluxes | Summing the above equation over all
the elements and using the identity
(REF ), we
obtain the following weak form in
terms of numerical fluxes:&\left( \mathbf {w}_1 ; \mu k_{1}^{-1} \mathbf {u}_{1} \right)
- \left(\mathrm {div}[\mathbf {w}_{1}] ; p_{1} \right)
+ \left(\lbrace \!\!\lbrace \mathbf {w}_{1} \rbrace \!\!\rb... | {
"cite_spans": []
} | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.032659754157066345,
0.06464800238609314,
-0.022526074200868607,
-0.03934432193636894,
-0.011484329588711262,
-0.011545375920832157,
0.032385047525167465,
0.05311025679111481,
0.03394172713160515,
0.040748387575149536,
-0.030034765601158142,
0.012949440628290176,
-0.01230082381516695,
-0... |
0cea57c11f16caaf2bfc9178f041bb6ffccd597b | subsection | 24 | 71 | Weak form in terms of numerical fluxes | That is,\llbracket p_1 \rrbracket = \mathbf {0}, \;
\llbracket p_2 \rrbracket = \mathbf {0}, \;
\llbracket \mathbf {u}_1 \rrbracket = 0
\; \mathrm {and} \;
\llbracket \mathbf {u}_2 \rrbracket = 0
\quad \mathrm {on} \; \Gamma ^{\mathrm {int}}Numerical fluxes are important components of DG
methods, which have to be selec... | {
"cite_spans": []
} | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.041470255702733994,
0.014052283018827438,
-0.04525414854288101,
-0.020048534497618675,
-0.002256223000586033,
-0.025510767474770546,
0.029676102101802826,
0.02430541440844536,
0.005637696478515863,
0.041134584695100784,
-0.027799412608146667,
-0.003944098949432373,
-0.026365194469690323,
... |
2a4cc71d61a91574c22283a80c14cc6e9e3f4bc9 | subsection | 25 | 71 | Weak form in terms of numerical fluxes | \\
{\mathop {p}^{\star }}_{2} &=
\left\lbrace \begin{array}{ll}
\lambda _{2}^{(1)} \lbrace \!\!\lbrace p_2\rbrace \!\!\rbrace
+ \frac{\lambda _{2}^{(2)}}{2} \llbracket p_2 \rrbracket \cdot \widehat{\mathbf {n}}
+ \lambda _{2}^{(3)} \llbracket \mathbf {u}_2 \rrbracket & \mathrm {on} \; \Gamma ^{\mathrm {int}} \\
p_2 \q... | {
"cite_spans": []
} | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.008193624205887318,
0.008300431072711945,
-0.023695899173617363,
-0.023070314899086952,
-0.008811579085886478,
-0.00744216050952673,
0.0023440325167030096,
0.07226257771253586,
0.0075527820736169815,
0.035887155681848526,
-0.031157132238149643,
0.022627828642725945,
-0.019393103197216988,... |
5cd102de4648b254416dc847959f4a8e4279ece1 | subsection | 26 | 71 | Weak form in terms of numerical fluxes | It is easy to check that these numerical
fluxes satisfy the following relations
on \Gamma ^{\mathrm {int}}:&\lbrace \!\!\lbrace \mathop {p_{1}}^{\star }\rbrace \!\!\rbrace
= \lambda _{1}^{(1)} \lbrace \!\!\lbrace p_1\rbrace \!\!\rbrace + \lambda _{1}^{(3)}
\llbracket \mathbf {u}_{1} \rrbracket \quad \mathrm {and} \qua... | {
"cite_spans": []
} | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.009246346540749073,
0.005798039026558399,
-0.011535046622157097,
-0.029722580686211586,
-0.014525613747537136,
-0.01969807595014572,
0.04900868982076645,
0.010573792271316051,
0.025038374587893486,
0.03564268350601196,
-0.011504529975354671,
-0.0012339905370026827,
-0.014525613747537136,
... |
3cf59c79db38de838ae19735e81edbf0f516d081 | subsection | 27 | 71 | The classical mixed DG formulation | This formulation is based on the Galerkin formalism
and can be obtained by making the following choices:\lambda _{1}^{(1)} = \lambda _{2}^{(1)}
= \Lambda _{1}^{(1)} = \Lambda _{2}^{(1)} = 1and the other constants in equations
(REF )–(REF )
are taken to be zeros.
The numerical fluxes on \Gamma ^{\mathrm {int}}
under the... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1006/jcph.1996.5572",
"end": 841,
"openalex_id": "https://openalex.org/W2079263908",
"raw": "F. Bassi and S. Rebay. A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier–Stokes eq... | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.00464327959343791,
-0.010057260282337666,
-0.026707442477345467,
-0.029378186911344528,
0.0038611332420259714,
-0.026142772287130356,
0.045326344668865204,
0.019259881228208542,
0.006566215772181749,
0.02539496310055256,
-0.031407952308654785,
0.020389225333929062,
-0.011049251072108746,
... |
c3c20c50ce99c1a94cb234848bdf074abc9c059a | subsection | 28 | 71 | The classical mixed DG formulation | The corresponding weak formulation reads: Find
\left(\mathbf {u}_1(\mathbf {x}),
\mathbf {u}_2(\mathbf {x})\right) \in \mathcal {U} \times \mathcal {U} ,
\left(p_1(\mathbf {x}), p_2(\mathbf {x})
\right) \in \mathcal {P} such that we
have\mathcal {B}^{\mathrm {DG}}_{\mathrm {Gal}}
(\mathbf {w}_1,\mathbf {w}_2,q_1,q_2;
\... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/978-1-4612-3172-1",
"end": 2614,
"openalex_id": "https://openalex.org/W3149667697",
"raw": "F. Brezzi and M. Fortin. Mixed and Hybrid Finite Element Methods, volume 15 of Springer series in computational mathematics. Springer-Verlag... | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.055411722511053085,
0.021740557625889778,
-0.03814132884144783,
-0.051963746547698975,
-0.003047492355108261,
-0.022716976702213287,
0.022137228399515152,
0.05187220871448517,
0.007864742539823055,
0.03914825990796089,
-0.006358159705996513,
-0.005675429943948984,
-0.009275971911847591,
... |
5244bd5fce6a724bd1f62d62a4aedf1f12338658 | subsection | 29 | 71 | The classical mixed DG formulation | Specifically,
equal-order interpolation for all the field
variables is not stable under the classical
mixed DG formulation.
This numerical instability (due to the interpolation
functions) is different from the aforementioned
instability due to the numerical fluxes (i.e.,
Bassi-Rebay DG method).
We develop a stabilized ... | {
"cite_spans": []
} | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.018297092989087105,
-0.04633025452494621,
-0.03589221090078354,
-0.00235771550796926,
-0.002519856207072735,
-0.022478412836790085,
0.03760136663913727,
0.021974824368953705,
0.017625639215111732,
0.029086120426654816,
-0.024996362626552582,
0.0110484529286623,
-0.0026018801145255566,
0... |
4a89c2ed83d10bb659647cccd537879580ae33fb | subsection | 30 | 71 | Proposed stabilized mixed DG formulation | This formulation makes the following choices:&\lambda _{1}^{(1)} = \lambda _{2}^{(1)} = 1, \;
\lambda _{1}^{(3)} = \eta _u h_{\Upsilon }
\lbrace \!\!\lbrace \mu k_1^{-1} \rbrace \!\!\rbrace
\; \mathrm {and} \;
\lambda _{2}^{(3)} = \eta _u h_{\Upsilon }
\lbrace \!\!\lbrace \mu k_2^{-1} \rbrace \!\!\rbrace \\
&\Lambda _... | {
"cite_spans": []
} | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.022539108991622925,
0.016755545511841774,
-0.00857615377753973,
-0.006924252025783062,
0.005249460693448782,
-0.0638480931520462,
0.053684890270233154,
0.02736128680408001,
0.030443819239735603,
0.004394897259771824,
-0.03659362345933914,
0.03174092248082161,
0.033602651208639145,
-0.01... |
55d4c71e44a3d154dbbd023adf36e898da8da3f2 | subsection | 31 | 71 | Proposed stabilized mixed DG formulation | The corresponding numerical fluxes
on \Gamma ^{\mathrm {int}} take the
following form:&\overset{*}{p}_{1} = \lbrace \!\!\lbrace p_{1}\rbrace \!\!\rbrace
+ \eta _{u} h_{\Upsilon } \lbrace \!\!\lbrace \mu k_1^{-1}\rbrace \!\!\rbrace
\llbracket \mathbf {u}_1 \rrbracket , \;
\overset{*}{p}_{2} = \lbrace \!\!\lbrace p_{2}... | {
"cite_spans": []
} | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.021273216232657433,
0.008118616417050362,
-0.03274915739893913,
-0.03369531035423279,
0.010163531638681889,
-0.054052893072366714,
0.025820862501859665,
0.029391832649707794,
0.019808202981948853,
0.052191104739904404,
-0.011002861894667149,
0.024355849251151085,
-0.02508835680782795,
0... |
0c1119d596d36d68d548b4f02724ca0ff64b2a19 | subsection | 32 | 71 | Proposed stabilized mixed DG formulation | We make the following
recommendation, which is based on the
theoretical convergence analysis (see
\mathsection \ref {Sec:S4_DG_Error})
and extensive numerical simulations
(see \mathsection \ref {Sec:S5_DG_Patch_tests}–\mathsection \ref {Sec:S8_DG_NR}):For conforming approximations, the
parameters can be taken to be \et... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1016/j.cma.2005.06.018",
"end": 897,
"openalex_id": "https://openalex.org/W1985247669",
"raw": "T. J. R. Hughes, A. Masud, and J. Wan. A stabilized mixed discontinuous Galerkin method for Darcy flow. Computer Methods in Applied Mechanics... | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.01274765282869339,
0.02898927591741085,
-0.039608508348464966,
0.007411337457597256,
0.01795809343457222,
-0.0057978555560112,
0.04140889272093773,
0.013174863532185555,
0.03948644548654556,
0.04757292941212654,
-0.019636420533061028,
0.025907259434461594,
-0.004874775651842356,
-0.0123... |
c2693f5c10f8989312458a816800a37d48d23953 | subsection | 33 | 71 | A THEORETICAL ANALYSIS OF THE PROPOSED DG FORMULATION | We start by grouping the field
variables and their corresponding
weighting functions as follows:\mathbf {U} &= (\mathbf {u}_1(\mathbf {x}),
\mathbf {u}_2(\mathbf {x}),p_1(\mathbf {x}),
p_2(\mathbf {x})) \in \mathbb {U} \\
\mathbf {W} &= (\mathbf {w}_1(\mathbf {x}),
\mathbf {w}_2(\mathbf {x}),q_1(\mathbf {x}),
q_2(\mat... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1016/j.cma.2018.04.004",
"end": 2386,
"openalex_id": "https://openalex.org/W2618072050",
"raw": "S. H. S. Joodat, K. B. Nakshatrala, and R. Ballarini. Modeling flow in porous media with double porosity/permeability: A stabilized mixed fo... | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.03609577938914299,
0.0036271223798394203,
-0.026377685368061066,
-0.034570176154375076,
0.00933669414371252,
-0.021312681958079338,
0.02779649756848812,
0.03588219732046127,
0.016323957592248917,
0.04534093663096428,
-0.007971278391778469,
0.029993366450071335,
-0.015568784438073635,
0.... |
f0b29c80305b6ae528d8b754e41ad8e419b81765 | subsection | 34 | 71 | Convergence theorem and error analysis | In order to perform the error analysis
of the proposed stabilized mixed DG
formulation, we need to define the
finite element solution \mathbf {U}^{h}
and the corresponding weighting function as&\mathbf {U}^{h} = (\mathbf {u}_{1}^{h}(\mathbf {x}),\mathbf {u}_{2}^{h}(\mathbf {x}),
p_{1}^{h}(\mathbf {x}),
p_{2}^{h}(\mathb... | {
"cite_spans": []
} | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
0.0005320971831679344,
-0.03960786387324333,
-0.05077618360519409,
-0.03832625225186348,
0.02110080048441887,
-0.035061199218034744,
0.06231067702174187,
0.029446525499224663,
-0.01576075702905655,
0.032131802290678024,
0.016645677387714386,
-0.0047678956761956215,
-0.015371696092188358,
0... |
42c74c0223bac5838cf2b420dec1ec2d7120d815 | subsection | 35 | 71 | Convergence theorem and error analysis | If we define \widetilde{\mathbf {U}}^{h} as an “interpolate” of \mathbf {U} onto \mathbb {U}^{h} , decomposition of the error can be performed as follows:\mathbf {E} := \mathbf {U}^{h} - \mathbf {U}
= \mathbf {E}^h + \mathbf {H}where \mathbf {E}^h = \mathbf {U}^{h} - \widetilde{\mathbf {U}}^{h} is the approximation err... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/978-1-4757-3658-8",
"end": 410,
"openalex_id": "https://openalex.org/W1492326914",
"raw": "S. C. Brenner and L. R. Scott. The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York, 1994.",
"source_ref_id": "... | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
0.002268014708533883,
-0.012833643704652786,
-0.014550391584634781,
0.040805187076330185,
0.013215143233537674,
-0.0310693196952343,
0.0557599700987339,
0.03189335763454437,
0.014222301542758942,
0.016251878812909126,
-0.00012720625090878457,
-0.008019119501113892,
0.0247516892850399,
0.02... |
63ae33b99c3e5c0c00579a7d0642b46490712259 | subsection | 36 | 71 | Convergence theorem and error analysis | The components of \mathbf {E} and \mathbf {H} are as follows:\mathbf {E} = \left\lbrace \mathbf {e}_{\mathbf {u}_{1}},\mathbf {e}_{\mathbf {u}_{2}},e_{p_{1}},e_{p_{2}} \right\rbrace
\quad \mathrm {and} \quad \mathbf {H} =
\left\lbrace {\eta }_{\mathbf {u}_{1}},
{\eta }_{\mathbf {u}_{2}},\eta _{p_{1}},\eta _{p_{2}}\rig... | {
"cite_spans": []
} | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
0.00690475944429636,
0.0006089349626563489,
-0.019577473402023315,
0.02240041457116604,
0.015228618867695332,
-0.07275556772947311,
0.03375321254134178,
0.008659560233354568,
-0.007351089268922806,
0.03302077203989029,
-0.045197565108537674,
0.02557431347668171,
0.01386292651295662,
0.0197... |
ae8f95061a15d1f90ce233d1b16425fc0316060a | subsection | 37 | 71 | Convergence theorem and error analysis | The Cauchy-Schwarz inequality implies
the following:\left\Vert \sqrt{h_{\Upsilon } \lbrace \!\!\lbrace \mu k_1^{-1} \rbrace \!\!\rbrace } \; \lbrace \!\!\lbrace
\mathbf {e}_{\mathbf {u}_1} \rbrace \!\!\rbrace
\right\Vert ^2_{\Upsilon }
&\le \frac{1}{2} \left(
\left\Vert \sqrt{h_{\Upsilon }
\lbrace \!\!\lbrace \mu k_1... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/978-1-4612-3172-1",
"end": 4732,
"openalex_id": "https://openalex.org/W3149667697",
"raw": "F. Brezzi and M. Fortin. Mixed and Hybrid Finite Element Methods, volume 15 of Springer series in computational mathematics. Springer-Verlag... | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.03819628432393074,
-0.007684256415814161,
-0.03779967501759529,
0.012874466367065907,
0.013782084919512272,
-0.012302436865866184,
0.020852364599704742,
-0.00013704861339647323,
0.01995237171649933,
0.013690561056137085,
-0.03545054420828819,
0.03493190556764603,
0.007825356908142567,
0... |
20de42ec9137d28c306e877322e76a0a68eb84cd | subsection | 38 | 71 | Convergence theorem and error analysis | A similar
definition holds for A \gtrsim B. The
notation A \sim B denotes the case when
A \lesssim B and A \gtrsim B hold
simultaneously.Lemma 4.3 (Estimates for interpolation
errors on \Gamma ^{\mathrm {int}}.)
If polynomial orders used for interpolation of
\mathbf {u}_1, \mathbf {u}_2, p_1 and p_2
are, respectively,... | {
"cite_spans": []
} | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.0030216870363801718,
-0.010751406662166119,
-0.030384741723537445,
0.021838556975126266,
0.01187309343367815,
-0.04837751388549805,
0.012964258901774883,
0.02487550489604473,
0.006337149068713188,
0.012483535334467888,
-0.026111649349331856,
0.008462250232696533,
0.026645785197615623,
0... |
a7fd8c91e91a255899b3831eeca86ef43b738b50 | subsection | 39 | 71 | Convergence theorem and error analysis | The boundedness of the drag coefficient
\mu /k_1(\mathbf {x}) and the linearity
of a norm imply the following:\left\Vert \sqrt{\frac{h_{\Upsilon }}{\eta _p}
\lbrace \!\!\lbrace \mu k_1^{-1} \rbrace \!\!\rbrace } \;
\lbrace \!\!\lbrace {\eta }_{\mathbf {u}_1} \rbrace \!\!\rbrace
\right\Vert ^2_{\Upsilon }
&\le \frac{1}... | {
"cite_spans": []
} | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.043967004865407944,
-0.0073837717063724995,
-0.024882089346647263,
-0.023798933252692223,
-0.0076927002519369125,
-0.02581268921494484,
-0.0015541771426796913,
-0.001216644188389182,
0.024760043248534203,
0.00791009422391653,
-0.04695712774991989,
0.023951491340994835,
0.01711692474782467... |
2744c1d4a7422ae5f5158a0596518286b151102d | subsection | 40 | 71 | Convergence theorem and error analysis | By reasoning out on similar lines, one can establish
the estimate (REF ).We now establish the estimate
(REF ). | {
"cite_spans": []
} | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.004999921191483736,
0.026162821799516678,
-0.006025839596986771,
0.018214816227555275,
-0.0071013374254107475,
-0.013554325327277184,
-0.008352271281182766,
0.030663132667541504,
0.047047313302755356,
0.015507002361118793,
-0.017162201926112175,
0.019404729828238487,
-0.01261612493544817,... |
df5caaec6eff595f8d9a6e8067cd6d99b2c1f98f | subsection | 41 | 71 | Convergence theorem and error analysis | The boundedness of the drag coefficient
\mu /k_1(\mathbf {x}) and the linearity
of a norm imply the following:\left\Vert \sqrt{h_{\Upsilon }^{-1}
\lbrace \!\!\lbrace \mu ^{-1} k_1\rbrace \!\!\rbrace } \;
\llbracket \eta _{p_1} \rrbracket \right\Vert ^2_{\Upsilon }
&\le \left(\inf _{\mathbf {x} \in \Omega }
\frac{\mu }{... | {
"cite_spans": []
} | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.04583762586116791,
-0.01480872742831707,
-0.02946486510336399,
-0.020828349515795708,
-0.0008063355926424265,
-0.016357501968741417,
0.005012536887079477,
0.006389649119228125,
0.01030736230313778,
0.023590203374624252,
-0.05313136428594589,
0.023422354832291603,
0.00475695077329874,
0.... |
f8f41c6bd57857800a39a644f052129eb005ddee | subsection | 42 | 71 | Convergence theorem and error analysis | By reasoning out on similar lines, one can establish
the estimate (REF ).Lemma 4.4 (Estimate for \mathbf {H} under the stability norm.)
If polynomial orders used for interpolation of
\mathbf {u}_1, \mathbf {u}_2, p_1 and p_2
are, respectively, p, q, r and s then
the following estimate holds:\left(\Vert \mathbf {H}\Ver... | {
"cite_spans": []
} | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
0.0020693750120699406,
0.0014685887144878507,
-0.023207517340779305,
-0.02236832305788994,
0.018660614266991615,
-0.04128832370042801,
0.030317775905132294,
-0.0031336250249296427,
0.01094003301113844,
0.02357371151447296,
-0.012061500921845436,
0.02145284041762352,
0.01049754861742258,
0.... |
32c38f945e5623b822c9c5443bae2abb3cc0dd65 | subsection | 43 | 71 | Convergence theorem and error analysis | Equation (REF ) implies
that for all \mathbf {W}^{h} \in \mathbb {U}^{h} \subset \mathbb {U} we have the following:\mathcal {B}_{\mathrm {stab}}^{\mathrm {DG}}(\mathbf {W}^{h};
\mathbf {U}^{h})
&= \mathcal {L}_{\mathrm {stab}}^{\mathrm {DG}}(\mathbf {W}^{h}) \\
\mathcal {B}_{\mathrm {stab}}^{\mathrm {DG}}(\mathbf {W}^{... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/bfb0082920",
"end": 975,
"openalex_id": "https://openalex.org/W3021722416",
"raw": "L. C. Evans. Partial Differential Equations. American Mathematical Society, Providence, Rhode Island, 1998.",
"source_ref_id": "104c775ce53825... | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.005400141701102257,
-0.0006740642711520195,
-0.02927364967763424,
-0.048814840614795685,
-0.011669797822833061,
0.016169914975762367,
0.032431360334157944,
0.015895333141088486,
0.01215031836181879,
0.02376672439277172,
-0.018641166388988495,
0.012691858224570751,
0.01807674579322338,
0... |
c9682cdb87ff39a1fdf1fc663918617f5b1f8059 | subsection | 44 | 71 | Convergence theorem and error analysis | We now expand \mathcal {B}_{\mathrm {stab}}^{\mathrm {DG}}
(\mathbf {H};\mathbf {E}) as follows:\mathcal {B}_{\mathrm {stab}}^{\mathrm {DG}}(\mathbf {H};\mathbf {E}) &=
\mathcal {B}_{\mathrm {stab}}^{\mathrm {DG}}(
{\eta }_{\mathbf {u}_{1}},{\eta }_{\mathbf {u}_{2}},
\eta _{p_1},\eta _{p_{2}};\mathbf {e}_{\mathbf {u}_{... | {
"cite_spans": []
} | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.057185932993888855,
0.015379780903458595,
0.005031233187764883,
-0.02139132283627987,
-0.008277312852442265,
-0.03594718873500824,
-0.011275454424321651,
0.018171943724155426,
0.019499365240335464,
0.043698109686374664,
-0.039914194494485855,
0.04189769923686981,
-0.0032212859950959682,
... |
bf856f7d0e3c17ad10cff42a925cd7984e5a40a2 | subsection | 45 | 71 | Convergence theorem and error analysis | (See the
online version for the colored text.)
The red-colored terms
contain interpolation errors and
contribute to
\Vert \mathbf {H}\Vert _{\mathrm {stab}}^{\mathrm {DG}}.
The blue-colored terms
contain approximation errors and
contribute to
\Vert \mathbf {E}\Vert _{\mathrm {stab}}^{\mathrm {DG}}. | {
"cite_spans": []
} | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.03653035685420036,
-0.036377765238285065,
-0.021454336121678352,
-0.006172317545861006,
-0.012832927517592907,
-0.03894129768013954,
0.009170737117528915,
0.03701864928007126,
0.0008301920024678111,
0.007274789735674858,
-0.011894491501152515,
-0.035309623926877975,
0.02703917771577835,
... |
bdac4ef0dbd2f92ca3389d5f26cf345820d85ceb | subsection | 46 | 71 | Convergence theorem and error analysis | We employ Lemma REF
on the magenta-colored
terms and employ Lemma REF
on the green-colored terms.:2 \mathcal {B}_{\mathrm {stab}}^{\mathrm {DG}}
(\mathbf {H};\mathbf {E})
&\le {\color {red}
\frac{\varepsilon _1}{2} \left\Vert \sqrt{\frac{\mu }{k_1}}
{\eta }_{\mathbf {u}_{1}} \right\Vert ^{2} }
{\color {blue} + \frac{... | {
"cite_spans": []
} | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.04274506866931915,
0.009427713230252266,
-0.025781286880373955,
-0.030434122309088707,
-0.016201021149754524,
-0.02149457484483719,
0.018962211906909943,
0.0034057232551276684,
0.03746676817536354,
0.03005274198949337,
-0.053789831697940826,
0.05854945629835129,
-0.004702415317296982,
0... |
4089571589ddd08ce61453ed46c2c85411f21406 | subsection | 47 | 71 | Convergence theorem and error analysis | After employing Lemma
REF ,
the above inequality can be grouped
as follows:2 \mathcal {B}_{\mathrm {stab}}^{\mathrm {DG}}
(\mathbf {H};\mathbf {E})
& \le {\color {blue}
\left(\frac{1}{2\varepsilon _1}
+ \frac{1}{2\varepsilon _3}
+ \frac{\mathcal {C}_{\mathbf {e}_{\mathbf {u}_1}}}{\varepsilon _{11}}
\right) \left\Vert \... | {
"cite_spans": []
} | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.028391370549798012,
0.020183656364679337,
-0.017956282943487167,
-0.035515911877155304,
0.006151972338557243,
0.009084933437407017,
-0.005942202638834715,
0.019725976511836052,
0.030832326039671898,
0.04106908664107323,
-0.040245264768600464,
0.016262870281934738,
-0.014012613333761692,
... |
c86bd3dbce15272b9a4ebbf6c8c745d40ebc5802 | subsection | 48 | 71 | Convergence theorem and error analysis | This can be achieved by choosing
these coefficients as follows:\frac{1}{2\varepsilon _1} + \frac{1}{2\varepsilon _3}
+ \frac{\mathcal {C}_{\mathbf {e}_{\mathbf {u}_1}}}{\varepsilon _{11}}
= \frac{1}{2\varepsilon _5} + \frac{1}{2\varepsilon _7}
+ \frac{\mathcal {C}_{\mathbf {e}_{\mathbf {u}_2}}}{\varepsilon _{13}}
= \fr... | {
"cite_spans": []
} | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.02336476556956768,
0.02211335487663746,
-0.0365656279027462,
-0.02606598287820816,
0.008683573454618454,
-0.034734293818473816,
0.020098887383937836,
0.02160973846912384,
-0.003971704747527838,
0.03050696663558483,
-0.05213196575641632,
0.012552266009151936,
0.0015585413202643394,
0.007... |
eb94b20d40f92b202d215110834fe51b500afdf7 | subsection | 49 | 71 | Convergence theorem and error analysis | \\
&\qquad \qquad \left. + h_{\omega }^{2r}|p_1|^2_{H^{r+1}(\omega )}
+ h_{\omega }^{2s}|p_2|^2_{H^{s+1}(\omega )} \right)As h \rightarrow 0, h_{\omega } \rightarrow 0 \;
\forall \omega \in \mathcal {T}_{h}, which
in turn implies that \Vert \mathbf {H}\Vert _{\mathrm {stab}}^{\mathrm {DG}}
\rightarrow 0 (using Lemma
RE... | {
"cite_spans": []
} | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.014240806922316551,
-0.01949625089764595,
-0.040426500141620636,
-0.0037527994718402624,
-0.009137914516031742,
0.0018125183414667845,
0.019740335643291473,
0.03700931742787361,
0.025781428441405296,
0.03076990507543087,
-0.049884773790836334,
0.02212016098201275,
-0.006483495235443115,
... |
8d8834ca7132b00082c6b5d963dd56d4d79ac719 | subsection | 50 | 71 | PATCH TESTS | Patch tests are generally used to indicate the quality of a finite element. Despite some debated mathematical controversies regarding the patch test, “the patch test is the most practically useful technique for assessing element behavior” as nicely pinpointed by .
In this section, different constant flow patch tests ar... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 264,
"openalex_id": "",
"raw": "T. J. R. Hughes. The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Dover Publications, Inc., New York, 2012.",
"source_ref_id": "0c86c2fa792d93cbafb3d1276b4afcdca31... | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.01663307473063469,
-0.03074830025434494,
-0.03915639594197273,
-0.01735028065741062,
0.010681791231036186,
-0.04922780022025108,
0.04370378702878952,
0.018815211951732635,
-0.0010252612410113215,
0.061313483864068985,
-0.029268108308315277,
-0.0014077075757086277,
-0.0022603434044867754,
... |
49b8614f69febb2660295eb006527cacf15fd262 | subsection | 51 | 71 | Velocity-driven patch test | In reality, heterogeneity of the material properties
is indispensable when it comes to porous domains. In
many geological systems, medium properties can vary
by many orders of magnitude and rapid changes may
occur over small spatial scales.
The aim of this boundary value problem is
to show that the proposed stabilized ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1016/j.cma.2005.06.018",
"end": 1360,
"openalex_id": "https://openalex.org/W1985247669",
"raw": "T. J. R. Hughes, A. Masud, and J. Wan. A stabilized mixed discontinuous Galerkin method for Darcy flow. Computer Methods in Applied Mechanic... | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.0383295901119709,
-0.021331515163183212,
-0.03033408522605896,
-0.021285738795995712,
0.021712979301810265,
-0.01719643548130989,
-0.0022792525123804808,
0.002113315276801586,
0.008933907374739647,
0.02098056674003601,
-0.03948924317955971,
-0.01248915959149599,
0.03448442369699478,
-0.... |
06b3d46ca15356a2345e89cac82e457156b5016d | subsection | 52 | 71 | Non-conforming discretization | One of the features of DG formulations is that the global error of the computation can be controlled by adjusting the numerical resolution in a selected set of the elements. Such a non-conforming discretization can be obtained in two ways : One can either modify the local order of the interpolation, or locally change t... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 537,
"openalex_id": "https://openalex.org/W1583515859",
"raw": "J. S. Hesthaven and T. Warburton. Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Springer Science & Business Media, 2007.",
"sour... | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
0.00033449273905716836,
-0.019393425434827805,
-0.02282656542956829,
-0.01239744946360588,
-0.022384071722626686,
-0.006049955729395151,
0.021422794088721275,
0.0053633274510502815,
0.022048387676477432,
0.047972407191991806,
-0.014999006874859333,
-0.0031775617972016335,
-0.0080564348027110... |
03208e9557cbf1f9cf214ce9be1e963ea4a4d295 | subsection | 53 | 71 | Non-conforming polynomial orders | Since the element communication under the DG formulations takes place through fluxes, each element can independently possess a desired order of interpolation. Hence, the DG methods can easily support the non-conforming polynomial orders (see , , ).In order to investigate the performance of our proposed stabilized mixed... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1137/s00361445023830",
"end": 248,
"openalex_id": "https://openalex.org/W2057858153",
"raw": "J. F. Remacle, J. E. Flaherty, and M. S. Shephard. An adaptive discontinuous Galerkin technique with an orthogonal basis applied to compressibl... | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.006921924650669098,
-0.01923646777868271,
-0.05009718984365463,
-0.023340042680501938,
0.004931156057864428,
-0.017466895282268524,
-0.0018134298734366894,
-0.021478941664099693,
-0.011044569313526154,
0.03092174418270588,
0.000281977845588699,
0.016231246292591095,
0.0029175053350627422,... |
876d2dbffbc1bbc989f1717eeb94d726374245b6 | subsection | 54 | 71 | Non-conforming polynomial orders | In the left half, third order interpolation polynomials are employed for velocities and pressures in each pore-network while in the right half, first-order interpolation polynomials are used.Smooth velocity profiles along the non-conforming edge (x = 0.5) are not achievable for a coarse
mesh (e.g., of size 10 x 10 elem... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1016/j.cma.2010.01.015",
"end": 2353,
"openalex_id": "https://openalex.org/W2169525875",
"raw": "S. Badia and R. Codina. Stabilized continuous and discontinuous Galerkin techniques for Darcy flow. Computer Methods in Applied Mechanics an... | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.007446265313774347,
-0.01573176123201847,
-0.042236194014549255,
-0.03219594061374664,
-0.017089787870645523,
0.0049857934936881065,
0.027557283639907837,
0.03729236125946045,
0.03086843155324459,
0.028060823678970337,
-0.03057851456105709,
0.005977611523121595,
0.019134460017085075,
-0... |
73ffc8c4ab035cc3cdae761d35da5690a85a3ba8 | subsection | 55 | 71 | Non-conforming polynomial orders | The exact and numerical solutions match which shows that the proposed stabilized DG formulation supports non-conforming order refinement.][Figure: Non-conforming polynomial orders: This figure shows the exact and numerical solutions for the velocity profiles within the domain. In the left half of the domain, third orde... | {
"cite_spans": []
} | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.026415592059493065,
-0.027621155604720116,
-0.04306458681821823,
-0.026583455502986908,
-0.010315967723727226,
-0.022692076861858368,
0.033969443291425705,
-0.00012470532965380698,
0.02870463766157627,
0.02238687127828598,
-0.013947920873761177,
0.0074012489058077335,
0.029894942417740822... |
af5a1e9f44774cf081f29508320ef9ba9db7b511 | subsection | 56 | 71 | Non-conforming element refinement | In mesh refinement procedures, one can either uphold the conformity of the mesh or produce irregular (non-conforming) meshes. The ability of DG formulations to support non-conforming elements obviates the user from propagating refinements beyond the desired elements
. The non-conforming meshes introduce hanging nodes o... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 268,
"openalex_id": "https://openalex.org/W1583515859",
"raw": "J. S. Hesthaven and T. Warburton. Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Springer Science & Business Media, 2007.",
"sour... | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.004623696208000183,
-0.03332723677158356,
-0.04504670202732086,
0.008682782761752605,
-0.010727585293352604,
0.029497044160962105,
0.031068796291947365,
0.02221815660595894,
0.02334737591445446,
0.04220839589834213,
-0.038149308413267136,
-0.010796254500746727,
0.003580312477424741,
0.0... |
54942e12b3c1ca5bba27dda17e83658d1025d292 | subsection | 57 | 71 | Non-conforming element refinement | REF shows the velocity and pressure profiles within the domain. Pressures in both pore-networks are varying linearly and velocities are constant throughout the domain. These results show that the proposed stabilized DG formulation is capable of handling non-conforming element refinement (with hanging nodes in the mesh)... | {
"cite_spans": []
} | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.036656737327575684,
-0.030933890491724014,
-0.0099577521905303,
-0.015642445534467697,
0.011041278019547462,
-0.011453323066234589,
0.025501001626253128,
0.05634332820773125,
0.012674196623265743,
0.05484775826334953,
-0.027484921738505363,
0.004933093208819628,
0.016374969854950905,
0.... |
393463f92c185f9c853e99494275d9ea78214feb | subsection | 58 | 71 | Non-constant Jacobian elements | In practice, many hydrogeological systems have complex shapes and modeling of such domains, especially in the 3D settings, requires using of elements with irregular shapes. Divergent boundaries in such elements result in non-constant Jacobian determinants.
Herein, the aim is to show that the proposed stabilized mixed D... | {
"cite_spans": []
} | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.0085659334436059,
-0.006388218142092228,
-0.0751788392663002,
-0.02045755460858345,
0.025720681995153427,
-0.02939724549651146,
0.02343236654996872,
-0.009664325043559074,
-0.010076222009956837,
0.0387183241546154,
-0.005568237975239754,
0.007536190561950207,
-0.002065206179395318,
0.01... |
927306b90c4e87541ad319b5689fcc0db32edaf7 | subsection | 59 | 71 | NUMERICAL CONVERGENCE ANALYSIS | In this section, we perform numerical convergence
analysis of the proposed stabilized DG formulation
with respect to both h- and p-refinements. | {
"cite_spans": []
} | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.03413916751742363,
-0.02381199225783348,
-0.011440739035606384,
-0.01099073700606823,
-0.016840768977999687,
0.008206823840737343,
0.044756174087524414,
0.014346687123179436,
0.04753245785832405,
0.056593526154756546,
-0.03490188345313072,
-0.009648356586694717,
0.038471393287181854,
0.... |
00e8369f1bce2f284d5731e5000594d51e0c8a3a | subsection | 60 | 71 | 2D numerical convergence analysis: | Convergence analysis in the 2D setting is performed on the boundary value problem described in Section REF . This problem was also employed by
for the convergence analysis
of the stabilized mixed continuous Galerkin (CG)
formulation of the DPP model. The exact solutions for the pressures and velocities are provided by... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1016/j.cma.2018.04.004",
"end": 251,
"openalex_id": "https://openalex.org/W2618072050",
"raw": "S. H. S. Joodat, K. B. Nakshatrala, and R. Ballarini. Modeling flow in porous media with double porosity/permeability: A stabilized mixed for... | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.04827269911766052,
-0.018796447664499283,
-0.04235303774476051,
-0.015455196611583233,
-0.015035632997751236,
0.03869139030575752,
0.010115296579897404,
0.0200780238956213,
0.016157012432813644,
0.048303209245204926,
-0.034358441829681396,
0.0018003091681748629,
0.06944921612739563,
0.0... |
47d492fe20c8cf28742765feaed8a81fd2075d08 | subsection | 61 | 71 | 3D numerical convergence analysis | The computational domain of this problem
is a unit cube with
pressure being prescribed on the entire
boundary of the two pore-networks. The
analytical solution takes the following
form:p_1(x,y,z) &= \frac{\mu }{\pi } \exp (\pi x)
\left(\sin (\pi y) + \sin (\pi z)\right)
- \frac{\mu }{\beta k_1} \left(\exp (\eta y)
+ \e... | {
"cite_spans": []
} | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.02896745689213276,
0.00794392079114914,
-0.061872538179159164,
-0.019978083670139313,
-0.005566849373281002,
0.024297256022691727,
0.03766685351729393,
0.016162559390068054,
0.049449190497398376,
0.04145185276865959,
-0.036171168088912964,
0.018696067854762077,
0.03266088664531708,
0.00... |
0170c23fe4104b6d59b28f8a76e75a512435236c | subsection | 62 | 71 | CANONICAL PROBLEM AND STRUCTURE PRESERVING PROPERTIES | In this section, first, robustness of the proposed stabilized mixed DG formulation is assessed using a standard test problem, with abrupt changes in material properties and elliptic singularities. In the literature, this problem is typically referred to as the quarter five-spot checkerboard problem. Second, the element... | {
"cite_spans": []
} | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
0.01641685701906681,
0.016798289492726326,
-0.01563873328268528,
-0.017698468640446663,
0.0038105109706521034,
-0.0521799698472023,
0.029751736670732498,
0.020658385008573532,
-0.0013578998623415828,
0.025449179112911224,
-0.020948274061083794,
-0.014799581840634346,
-0.017210235819220543,
... |
0f137dd9ba7384f33b2a41dfae0afd1c88e351a7 | subsection | 63 | 71 | Quarter five-spot checkerboard problem | The original form of this problem, known as “five-spot problem” with homogeneous properties, has been firstly designed for the Darcy equations. Herein, we extend this problem to the DPP model with modified boundary conditions and heterogeneous medium properties.
Fig. REF shows the computational domain and the boundary... | {
"cite_spans": []
} | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
0.003173181554302573,
0.0031102520879358053,
-0.05241851881146431,
0.03783408924937248,
0.01760505512356758,
-0.0333184078335762,
-0.02294454351067543,
0.027658550068736076,
0.022730965167284012,
0.044058408588171005,
-0.05263210088014603,
-0.00839062500745058,
0.03255562484264374,
-0.0172... |
7b351082545a75207ebd9f620d9ced145a11767a | subsection | 64 | 71 | Quarter five-spot checkerboard problem | Herein, we assume that sub-regions I and IV are more permeable compared to sub-regions II and III with the following drag coefficients:&\left(\frac{\mu }{k_1}\right)_I
= \left(\frac{\mu }{k_1}\right)_{IV} = 1,
\quad \left(\frac{\mu }{k_1}\right)_{II}
= \left(\frac{\mu }{k_1}\right)_{III} = 100,
\\
&\left(\frac{\mu }{k_... | {
"cite_spans": []
} | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
0.009699449874460697,
0.011660708114504814,
-0.047039661556482315,
-0.022787220776081085,
0.03217378631234169,
-0.04359028860926628,
0.003014384536072612,
0.017384221777319908,
0.053358420729637146,
0.025412406772375107,
0.009775763377547264,
-0.00507103418931365,
0.012858830392360687,
-0.... |
5f7fc0e7666bab7cca96042cd5d55baaf56929a9 | subsection | 65 | 71 | Element-wise mass balance | A DG method, when designed properly, can exhibit
superior element-wise properties compared to its
continuous counterpart. CG formulations may suffer from
poor element-wise conservation;
however, they satisfy a global mass balance
.
The importance of element-wise mass
balance in subsurface modeling is
discussed in , whi... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1006/jcph.2000.6577",
"end": 231,
"openalex_id": "https://openalex.org/W2083510900",
"raw": "T. J. R. Hughes, G. Engel, L. Mazzei, and M. G. Larson. The continuous Galerkin method is locally conservative. Journal of Computational Physics... | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.009740845300257206,
0.00992391537874937,
-0.039237987250089645,
-0.002486699726432562,
0.014012476429343224,
-0.01908503845334053,
0.027628302574157715,
0.010068845935165882,
-0.010023077949881554,
0.05174776166677475,
-0.011823265813291073,
0.007368564140051603,
0.006872749887406826,
0... |
43c7e3b6ce88dcf97d147b2a0ae544be60bf10a2 | subsection | 66 | 71 | COUPLED PROBLEM WITH HETEROGENEOUS MEDIUM PROPERTIES | In the previous sections, we used
patch tests and canonical problems
to demonstrate that the proposed
stabilized mixed DG formulation
can accurately capture the jumps
in the solution fields across
material interfaces.
We will further illustrate the
performance of this formulation
using a representative problem
pertaini... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1017/cbo9780511809064",
"end": 456,
"openalex_id": "https://openalex.org/W1433127933",
"raw": "P. G. Drazin. Introduction to Hydrodynamic Stability. Cambridge University Press, Cambridge, U.K., 2002.",
"source_ref_id": "b0222973eea... | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.02768339402973652,
-0.017092281952500343,
-0.0424865297973156,
0.009950150735676289,
0.03021671436727047,
-0.002703098813071847,
0.006581292022019625,
0.031162893399596214,
-0.022830406203866005,
0.056282445788383484,
-0.015840884298086166,
0.024905897676944733,
0.04254757612943649,
-0.... |
334f626f5017b47c3f5037fde9edc67ee48d3720 | subsection | 67 | 71 | COUPLED PROBLEM WITH HETEROGENEOUS MEDIUM PROPERTIES | Flow under the DPP model is governed by equations (REF )–()
and the transient advection-diffusion problem is governed by the following set of equations:&\frac{\partial c(\mathbf {x},t)}{\partial t} + \mathrm {div}\left[\mathbf {u}(\mathbf {x},t) c(\mathbf {x},t)- D(\mathbf {x},t) \mathrm {grad}[c(\mathbf {x},t)]\right]... | {
"cite_spans": []
} | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.04100906103849411,
-0.015698781237006187,
-0.03127551078796387,
0.003819807665422559,
-0.009695407934486866,
0.001755433389917016,
-0.028086934238672256,
0.013730712234973907,
0.011274440214037895,
0.04357212409377098,
-0.030741538852453232,
-0.021282603964209557,
0.02755296230316162,
0... |
131ce99a150a647755d5c1edb2e5626ecb16046d | subsection | 68 | 71 | COUPLED PROBLEM WITH HETEROGENEOUS MEDIUM PROPERTIES | Such heterogeneity in the permeability imposes a perturbation on the interface of the two fluids which causes the appearance of unstable finger-like patterns throughout the domain at the fluid-fluid interface. Moreover, a random function is used for defining the initial condition for the transport problem within the do... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1016/0045-7825(82)90071-8",
"end": 567,
"openalex_id": "https://openalex.org/W2073897969",
"raw": "A. N. Brooks and T. J. R. Hughes. Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis o... | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.02040100283920765,
-0.01652527041733265,
-0.0364685095846653,
-0.005523682106286287,
0.0269775427877903,
0.004592895973473787,
0.0006723404512740672,
0.0453796423971653,
-0.004482269752770662,
0.0404968298971653,
-0.009590149857103825,
0.0374145545065403,
0.0333862341940403,
-0.03729248... |
78e3c2fca82df7543a43b0a8c3e19c1714ab9384 | subsection | 69 | 71 | CONCLUDING REMARKS | A new stabilized mixed DG formulation has been
presented for the DPP mathematical model, which
describes the flow of a single-phase incompressible
fluid through a porous medium with two dominant pore-networks.
Some of the main findings of this paper
on the computational front and
the nature of flow through porous
media... | {
"cite_spans": []
} | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.004295836668461561,
-0.01473403349518776,
-0.06040114536881447,
0.008782430551946163,
0.010224550031125546,
0.006821453105658293,
-0.00842380803078413,
0.018831484019756317,
0.005028341896831989,
0.032749079167842865,
-0.031055161729454994,
0.00415849220007658,
0.022997606545686722,
0.0... |
5e7c126d05180e1da5759607630f3287226a9248 | subsection | 70 | 71 | COMPUTER IMPLEMENTATION | The numerical results pertaining to the non-conforming
discretization (Section REF ) and
non-constant Jacobian elements (Section REF ),
have been obtained using COMSOL Java API .
The numerical simulations for the 3D numerical convergence analysis (Section REF ) and the coupled problem
(Section ) were carried out using ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 178,
"openalex_id": "",
"raw": "COMSOL Java API Reference Guide, Version 4.3. COMSOL, Inc., Burlington, Massachusetts, www.comsol.com, 2012.",
"source_ref_id": "6ba62e54885d0784f274432561a36410c1aadd3a",
"start": 0
... | 10.1016/j.cma.2019.04.010 | 1805.01389 | A stabilized mixed discontinuous Galerkin formulation for double
porosity/permeability model | [
"M. S. Joshaghani",
"S. H. S. Joodat",
"K. B. Nakshatrala"
] | [
"cs.CE",
"cs.NA"
] | 2,018 | en | Computer Science | [
-0.03432363644242287,
-0.015483774244785309,
-0.02481980435550213,
-0.04899889975786209,
-0.007764769718050957,
-0.02184508927166462,
0.04448343440890312,
-0.007261356338858604,
-0.017634721472859383,
0.055680569261312485,
-0.04957858845591545,
0.030098017305135727,
-0.02657412365078926,
0... |
b5983f8827276d0bc30afa4dd1e0d4b782b189f7 | abstract | 0 | 10 | Abstract | Computer aided diagnosis (CAD) systems are designed to assist clinicians in
various tasks, including highlighting abnormal regions in a medical image. A
common approach consists in training a voxel-level binary classifier on a set
of feature vectors extracted from normal and pathological areas in patients'
scans. Howev... | {
"cite_spans": []
} | 1805.01717 | Feature extraction with regularized siamese networks for outlier
detection: application to lesion screening in medical imaging | [
"Z. Alaverdyan",
"C. Lartizien"
] | [
"cs.CV"
] | 2,018 | en | Computer Science | [
-0.024896245449781418,
-0.03795456886291504,
-0.02154013328254223,
-0.016353415325284004,
0.004870175383985043,
-0.023279208689928055,
0.034445907920598984,
0.02956429123878479,
0.05281298980116844,
0.0405479297041893,
0.03618498146533966,
0.031577955931425095,
-0.002126172883436084,
0.042... | |
4497180616f2796d673265e73008de678f56d393 | subsection | 1 | 10 | Introduction | Medically refractory epilepsy is often associated with malformations of cortical development. Among others, focal cortical dysplasia (FCD) is the second/third most common cause of medically intractable seizures in adults . For patients diagnosed with medically refractory epilepsy, the surgical removal of the lesions ma... | {
"cite_spans": []
} | 1805.01717 | Feature extraction with regularized siamese networks for outlier
detection: application to lesion screening in medical imaging | [
"Z. Alaverdyan",
"C. Lartizien"
] | [
"cs.CV"
] | 2,018 | en | Computer Science | [
-0.04104194417595863,
-0.045588597655296326,
-0.02033788524568081,
-0.023328302428126335,
0.013403475284576416,
-0.02270275540649891,
0.003505348227918148,
0.03457288071513176,
0.03673940524458885,
0.026364490389823914,
-0.015302999876439571,
0.019498737528920174,
-0.006110520102083683,
0.... | |
0e8fdd5c14f09995e197337e58c1fb5a33aa8435 | subsection | 2 | 10 | Method | In this study we propose to use a siamese network to learn patch-level representations in the context of outlier detection. Such an approach is applicable in cases where pathological samples are not available or their number is insufficient to adequately represent the nature of outliers.The motivation behind the archit... | {
"cite_spans": []
} | 1805.01717 | Feature extraction with regularized siamese networks for outlier
detection: application to lesion screening in medical imaging | [
"Z. Alaverdyan",
"C. Lartizien"
] | [
"cs.CV"
] | 2,018 | en | Computer Science | [
-0.011285360902547836,
-0.02812565304338932,
-0.022280767560005188,
0.0294380821287632,
-0.025622881948947906,
-0.032810717821121216,
0.026965832337737083,
0.021822942420840263,
0.05713643133640289,
0.03827408328652382,
0.017595700919628143,
0.03638174384832382,
-0.026050183922052383,
0.02... | |
9bdb8ab6649cbe2c649a9ff19e14283eb36f2118 | subsection | 3 | 10 | Architecture | We propose to use a stacked convolutional autoencoder to learn patch-level representations. An autoencoder
The proposed architecture is illustrated on figure . Our regularized siamese neural network (rSNN) consists of two identical (same architecture, shared parameters) subnetworks - stacked denoising autoencoders (sDA... | {
"cite_spans": []
} | 1805.01717 | Feature extraction with regularized siamese networks for outlier
detection: application to lesion screening in medical imaging | [
"Z. Alaverdyan",
"C. Lartizien"
] | [
"cs.CV"
] | 2,018 | en | Computer Science | [
-0.023041293025016785,
-0.030319901183247566,
-0.04367164522409439,
0.004852404817938805,
-0.008743484504520893,
-0.01290922798216343,
0.04837145656347275,
0.020615091547369957,
0.036927107721567154,
0.0634169653058052,
-0.004131410736590624,
-0.0035248601343482733,
-0.022736109793186188,
... | |
4c1f8717c967de62b0b01e35e85a7bd6c2518ce7 | subsection | 4 | 10 | Loss function | Our loss function is designed to maximize the cosine similarity between g(\mathbf {x_{1}}) and g(\mathbf {x_{2}}). In the absence of dissimilar pairs (the notion of dissimilar patches is not defined in our context) it is necessary to add a regularizing term. To this end we propose to use the mean squared error between ... | {
"cite_spans": []
} | 1805.01717 | Feature extraction with regularized siamese networks for outlier
detection: application to lesion screening in medical imaging | [
"Z. Alaverdyan",
"C. Lartizien"
] | [
"cs.CV"
] | 2,018 | en | Computer Science | [
-0.05354321748018265,
-0.023034261539578438,
-0.0762113705277443,
-0.008923869580030441,
0.03539038822054863,
0.0005563117447309196,
-0.0024655049201101065,
0.028724180534482002,
-0.029669959098100662,
0.0046869381330907345,
-0.035603951662778854,
0.0033102217130362988,
-0.01902233250439167,... | |
fc6b667a4ef96696ea2db714192a663e86b320de | subsection | 5 | 10 | Body | Your paper should not exceed 4 pages, including tables and
figures. It should consist of 2 columns each measuring 88mm wide,
with a gap of 6mm between the columns. We advise you to use the
gretsi.cls 2\epsilon class file to perform automatic page
setting :\documentclass{gretsi}In your file preamble, you have to enter t... | {
"cite_spans": []
} | 1805.01717 | Feature extraction with regularized siamese networks for outlier
detection: application to lesion screening in medical imaging | [
"Z. Alaverdyan",
"C. Lartizien"
] | [
"cs.CV"
] | 2,018 | en | Computer Science | [
-0.058911826461553574,
-0.005315037444233894,
-0.032355718314647675,
0.011660268530249596,
0.011087938211858273,
-0.02068018727004528,
0.033240921795368195,
-0.09535779803991318,
0.032111525535583496,
0.0033614845015108585,
-0.03165366128087044,
0.0641009509563446,
0.01239285059273243,
-0.... | |
0f2b5cdd8068cf71d4f7c7f5fe0130ec92cc2e1e | subsection | 6 | 10 | Section and subsection | This example file uses \section and \subsection. For
lower level sectioning commands, you obtain : | {
"cite_spans": []
} | 1805.01717 | Feature extraction with regularized siamese networks for outlier
detection: application to lesion screening in medical imaging | [
"Z. Alaverdyan",
"C. Lartizien"
] | [
"cs.CV"
] | 2,018 | en | Computer Science | [
-0.04236053302884102,
-0.0053294007666409016,
-0.03564633056521416,
0.04843384027481079,
0.005684185773134232,
-0.00646623782813549,
0.03537165746092796,
0.010536725632846355,
-0.01826569065451622,
-0.0029660766012966633,
-0.012085570022463799,
-0.0037157025653868914,
0.011391260661184788,
... | |
fba88ddc0f27678559593f0f98ffdea8040162d8 | subsection | 7 | 10 | Subsubsection | By means of \subsubsection. | {
"cite_spans": []
} | 1805.01717 | Feature extraction with regularized siamese networks for outlier
detection: application to lesion screening in medical imaging | [
"Z. Alaverdyan",
"C. Lartizien"
] | [
"cs.CV"
] | 2,018 | en | Computer Science | [
-0.061181798577308655,
0.0033852148335427046,
-0.014433411881327629,
0.01919369213283062,
-0.019071632996201515,
0.016615206375718117,
0.041042156517505646,
-0.027493666857481003,
-0.006400440353900194,
-0.002456426387652755,
-0.002048293361440301,
-0.005752004683017731,
-0.01554719544947147... | |
872d181be572c2b7cd02a4a6ee4e821e04ab7370 | subsection | 8 | 10 | Subsubsubsection | By means of \paragraph. | {
"cite_spans": []
} | 1805.01717 | Feature extraction with regularized siamese networks for outlier
detection: application to lesion screening in medical imaging | [
"Z. Alaverdyan",
"C. Lartizien"
] | [
"cs.CV"
] | 2,018 | en | Computer Science | [
-0.0010063049849122763,
0.02689076028764248,
-0.003895497415214777,
0.007943609729409218,
-0.03748223930597305,
-0.00852354709059,
0.03998512402176857,
-0.006024476140737534,
0.0009681512601673603,
-0.014841807074844837,
-0.012308398261666298,
-0.009660528041422367,
-0.014887590892612934,
... | |
44caeaf446031f6a5530858a10662ee746a308fe | subsection | 9 | 10 | Tables, figures and mathematics | The title of tables should appear at the top, as in table
REF .
[Table: 2 to the power]Captions should appear below graphical objects, as in figure REF .
[Figure: a square in an oval]Including Postscript graphics files is easily performed by
means of graphics, graphicx or epsfig
packages. To insert fig.eps file, with a... | {
"cite_spans": []
} | 1805.01717 | Feature extraction with regularized siamese networks for outlier
detection: application to lesion screening in medical imaging | [
"Z. Alaverdyan",
"C. Lartizien"
] | [
"cs.CV"
] | 2,018 | en | Computer Science | [
-0.04769265651702881,
-0.012960624881088734,
0.017407972365617752,
-0.009832987561821938,
0.013601409271359444,
-0.03387002646923065,
0.04998117312788963,
0.003814192023128271,
0.016050120815634727,
0.017697852104902267,
-0.019864313304424286,
0.019437123090028763,
-0.03750113770365715,
0.... | |
5d6b7b9cd0221140a3e898a23946738780fd377e | abstract | 0 | 80 | Abstract | A mathematical model for an elastoplastic porous continuum subject to large
strains in combination with reversible damage (aging), evolving porosity, water
and heat transfer is advanced. The inelastic response is modeled within the
frame of plasticity for nonsimple materials. Water and heat diffuse through the
continuu... | {
"cite_spans": []
} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.03546234965324402,
-0.013107641600072384,
-0.01773117482662201,
0.0031987070105969906,
0.01202423870563507,
-0.03982647508382797,
-0.023895885795354843,
0.009018179029226303,
-0.03430265188217163,
0.030091116204857826,
-0.016418885439634323,
0.033844877034425735,
-0.0059014903381466866,
... | |
26080915679ea45c7468c62ff5dc9a315f59099e | subsection | 1 | 80 | Introduction | The global movement of
tectonic plates in the upper lithospheric mantle originates
tectonic earthquakes. These occur on fault zones,
which are relatively
localized regions of partly damaged rocks with weakened elastic properties and
weakened shear-stress resistance. Tectonic earthquakes are very
complex thermomechanica... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 689,
"openalex_id": "",
"raw": "M. Cocco et al., The L'Aquila trial, in Geoethics: The Role and Responsibility of Geoscientists, S. Peppoloni and G. Di Capua, eds., Geological Society, London, 2015, pp. 43–55.",
"source_ref_... | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.011658642441034317,
0.040957849472761154,
-0.022065965458750725,
0.011155062355101109,
0.028185226023197174,
-0.060856893658638,
0.005852211266756058,
0.04971709102392197,
0.0036757532507181168,
0.01730484329164028,
0.0023824679665267467,
0.03290056809782982,
-0.061009492725133896,
0.00... | |
05b51c6501e55731229a21ecc62d1c9514f0ff72 | subsection | 2 | 80 | Introduction | Toupin
). Such materials are also known as weakly
nonlocal. Nonlocal-material concepts have the capacity to be fitted
with dispersion of elastic waves in general, cf. for a
thorough discussion.
This effectively entails the control the scale of
the damage and core regions. Eventually, the distinguished variational
struc... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.14311/610",
"end": 194,
"openalex_id": "https://openalex.org/W2621860062",
"raw": "M. Jirásek, Nonlocal theories in continuum mechanics, Acta Polytechnica, 44 (2004), pp. 16–34.",
"source_ref_id": "9d957b89e5cbfcaff5a8d171e1668a884... | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.01097050216048956,
0.00561494380235672,
-0.010245746932923794,
0.009993989951908588,
0.02535879611968994,
-0.059597745537757874,
0.033903274685144424,
-0.0016717045800760388,
-0.002321123145520687,
0.058712784200906754,
0.016509154811501503,
0.02612169459462166,
-0.03167560696601868,
0.... | |
94d81caa62c41791813fbf8f33061d638be734c8 | subsection | 3 | 80 | Thermodynamical modeling | We devote this section to present our general model
for damageable poroelastic continua with water and heat transfer. This
is formulated in Lagrangian coordinates with \Omega \subset \mathbb {R}^d (d=2 or
3) being a bounded smooth reference (fixed) configuration.
The variables of the model are&y:\Omega \rightarrow \mat... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/0-387-27649-1",
"end": 1631,
"openalex_id": "https://openalex.org/W1508861003",
"raw": "S. S. Antman, Nonlinear Problems of Elasticity, Springer, New York, 2nd ed., 2005.",
"source_ref_id": "fe136d7c647575e1fc5f5ba000d1ad42629... | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.026294535025954247,
0.02116687037050724,
-0.053596287965774536,
0.001566150109283626,
0.026889709755778313,
-0.04770557954907417,
0.0020525914151221514,
0.00774490786716342,
-0.0014211714733392,
0.026309795677661896,
0.01225450448691845,
0.046942535787820816,
0.007588483393192291,
0.008... | |
92812768cdbe9696aea5ba9046c2fa3746c6c868 | subsection | 4 | 80 | Small-strain mechanical stored energy | A crucial novelty of the present modelization is that of dealing with
finite strains. In order to motivate our assumptions on the mechanical stored
energy in the coming Subsection REF , let us comment on a classical
choice in the small-strain regime, namely\frac{1}{2}\lambda (\alpha )
I_1^2
+G(\alpha )I_2
-\gamma (\alp... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1063/1.1712886",
"end": 1369,
"openalex_id": "https://openalex.org/W2055072688",
"raw": "M. A. Biot, General theory of three-dimensional consolidation, J. Appl. Phys., 12 (1941), pp. 155–164.",
"source_ref_id": "e19638fa206de69a5a3... | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.021640073508024216,
0.016298729926347733,
-0.028568560257554054,
-0.007889928296208382,
0.020938068628311157,
-0.02307460643351078,
0.03864080831408501,
0.03452034294605255,
0.026142064481973648,
0.023166172206401825,
-0.015589093789458275,
0.010812406428158283,
-0.016588687896728516,
-... | |
980a7709dd29902f5097b9d15f8d48f217c2e832 | subsection | 5 | 80 | Small-strain mechanical stored energy | Indeed, we will replace
the term \gamma (\alpha )I_1\sqrt{I_2} by a bounded term
\gamma (\alpha )I_1\sqrt{I_2}/(1{+}\epsilon I_2) with a small, user-defined
parameter \epsilon >0.In the small-strain setting, the following additive decomposition of
the total small strain is often considerede(u)=e_{\rm el}+e_{\rm pl}
-\s... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1111/j.1365-246x.2004.02172.x",
"end": 528,
"openalex_id": "https://openalex.org/W2127394467",
"raw": "Y. Hamiel, V. Lyakhovsky, and A. Agnon, Coupled evolution of damage and porosity in poroelastic media: Theory and applications to defo... | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
0.0014536348171532154,
-0.011094934307038784,
-0.020007509738206863,
-0.014200600795447826,
0.008416583761572838,
-0.07758823782205582,
-0.02441801317036152,
0.007359741721302271,
0.023578643798828125,
0.04367772117257118,
-0.015978537499904633,
0.04914125055074692,
-0.023395508527755737,
... | |
0fb19df7c9c5b77471f940ce8c37cb58317713b3 | subsection | 6 | 80 | Mechanical stored energy | A focal point of our model is to move from the small to the finite strain
situation. In particular, by replacing the small strain e_{\rm el} with the
elastic Green-Lagrange strain
E_{\rm el}=\frac{1}{2}(F_{\rm el}^\top F_{\rm el}^{}-\mathbb {I}), we correspondingly
consider the mechanical stored energy (compare with (R... | {
"cite_spans": []
} | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
0.01326554175466299,
-0.034205112606287,
-0.029673926532268524,
-0.015470108948647976,
-0.0019699977710843086,
-0.03774462267756462,
0.02344926819205284,
0.03512050211429596,
0.012258611619472504,
0.003039861097931862,
-0.012670537456870079,
0.032770998775959015,
-0.00012199254706501961,
0... | |
fbc19278fe96e950ba4953da041495a7f914243e | subsection | 7 | 80 | Mechanical stored energy | Henceforth, we will however stick with a
small but fixed \epsilon >0 in (REF ). This yields a 3rd-order
polynomial growth of \psi _{_{\rm M}} with respect to F_{\rm el}, which in turn
ensures that its derivative has a 2nd-order polynomial
growth. In
particular, all driving forces of the system, to be
defined in (REF b,... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1111/j.1365-246x.2004.02172.x",
"end": 1166,
"openalex_id": "https://openalex.org/W2127394467",
"raw": "Y. Hamiel, V. Lyakhovsky, and A. Agnon, Coupled evolution of damage and porosity in poroelastic media: Theory and applications to def... | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.04007893428206444,
0.021000996232032776,
-0.03250880911946297,
-0.02419082634150982,
-0.022023573517799377,
-0.05802746117115021,
0.022176196798682213,
0.022863002493977547,
0.01405662577599287,
0.013667435385286808,
-0.025884948670864105,
0.04542075842618942,
-0.006066784728318453,
0.0... | |
7c491cd1de1873b85f32260795e04726a2c111a7 | subsection | 8 | 80 | Mechanical stored energy | As \psi _{_{\rm M}} depends on the elastic
Cauchy-Green tensor F_{\rm el}^\top F_{\rm el} and
^\top rather than on F_{\rm el} and , so that
the mechanical energy is both frame- and plastic-indifferent, namely&\forall R_1,{\color {black}R_2}\in {\rm SO}(d):\ \ \psi _{_{\rm M}}(R_1F_{\rm el}{\color {black},R_2},\alpha ,\... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/bf00281393",
"end": 1004,
"openalex_id": "https://openalex.org/W2078419759",
"raw": "E. Kröner, Allgemeine Kontinuumstheorie der Versetzungen und Eigenspannungen, Arch. Ration. Mech. Anal., 4 (1960), pp. 273–334.",
"source_ref... | 1807.00910 | Thermodynamics of elastoplastic porous rocks at large strains towards
earthquake modeling | [
"Tomas Roubicek",
"Ulisse Stefanelli"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.02408076450228691,
-0.011849628761410713,
-0.023195667192339897,
-0.024981122463941574,
0.0194263719022274,
-0.023760298267006874,
0.003937159199267626,
0.05820281058549881,
0.02192906104028225,
0.030810561031103134,
-0.0194416306912899,
0.014512551948428154,
0.02163911610841751,
-0.013... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.