$$ \begin{align} \Delta U(\mathbf{x}) &= f(\mathbf{x}) \quad \text{in } \Omega \ U(\mathbf{x}) &= U_0(\mathbf{x}) \quad \text{on } \partial\Omega \tag{1} \end{align} where **x** are the Cartesian coordinates, Ω is a Lipschitz domain with boundary ∂Ω, f(**x**) ∈ L²(Ω) : Ω ↦ R is a given source term, and U(**x**): Ω ↦ R is the unknown solution. According to a classical variational approach, we seek for a solution $U \in H^1(\Omega)$, such as $U(\mathbf{x}) = U_0(\mathbf{x})$ on $\partial\Omega$ and: \int_{\Omega} \nabla (\nabla U(\mathbf{x})) \psi(\mathbf{x}) d\Omega = \int_{\Omega} f(\mathbf{x}) \psi(\mathbf{x}) d\Omega \quad \forall \psi \in H_{\partial\Omega_p}^{1}(\Omega), where $\psi(\mathbf{x})$ are test functions. After integrating by parts and using boundary conditions, we obtain: -\int_{\Omega} \nabla U(\mathbf{x}) \nabla \psi(\mathbf{x}) d\Omega = \int_{\Omega} f(\mathbf{x}) \psi(\mathbf{x}) d\Omega. \quad (2) U(\mathbf{x}) = T(\xi, \eta) = \sum_{i=1}^{n_i} \sum_{j=1}^{n_j} \hat{N}i^{p_i}(\xi) \hat{N}j^{p_j}(\eta) T{ij}, where $\hat{N}_i$ functions are B-Spline basis functions and $\mathbf{u} = (\xi, \eta) \in \mathcal{P}$ are domain parameters. Then, we define the test functions $\psi(\mathbf{x})$ in the physical domain such as: N{ij}(\mathbf{x}) = N_{ij}(x, y) = N_{ij}(\mathcal{T}(\xi, \eta)) = \hat{N}{ij}(\xi, \eta) = \hat{N}{i}^{p_i}(\xi) \hat{N}{j}^{p_j}(\eta). \sum{k=1}^{n_k} \sum_{l=1}^{n_l} T_{kl} \int_{\Omega} \nabla N_{kl}(\mathbf{x}) \nabla N_{ij}(\mathbf{x}) d\Omega = \int_{\Omega} f(\mathbf{x}) N_{ij}(\mathbf{x}) d\Omega. M_{ij,kl} = \int_{\Omega} \nabla N_{kl}(\mathbf{x}) \nabla N_{ij}(\mathbf{x}) d\Omega \ = \int_{\mathcal{P}} \nabla_u \tilde{N}{kl}(\mathbf{u}) B(\mathbf{u})^T B(\mathbf{u}) \nabla_u \tilde{N}{kl}(\mathbf{u}) J(\mathbf{u}) d\mathcal{P} $$
where J is the Jacobian of the transformation, B**^T is the transposed of the inverse of the Jacobian matrix. The above integrations are performed in the parameter space using classical Gauss quadrature rules.
Starting from a planar B-spline surface as computational domain, an isogeometric solver for thermal conduction problem (1) has been implemented in the AXEL¹ platform, yielding a B-spline surface as solution field. Gauss-Seidel algorithm is employed to solve the linear system. Fig.1 shows an example of planar B-spline surface as computational domain and the corresponding isogeometric analysis results for two-dimensional heat conduction problems.
In order to improve the simulation results, refinement operation can be performed for two parametric directions. There are three kinds of refinement operations in isogeometric analysis: h-refinement by knot insertion, p-refinement by
Fig.1 An example of isogeometric analysis for two- dimensional heat conduction problem: (a) computational domain with control points and iso-parametric curves; (b) isogeometric simulation result.
Fig.2 Comparison of three kinds of refinement methods for the computational domain in Fig.1: (a) h-refinement; (b) p-refinement; (c) k-refinement.