| """Independent conforming P1 finite-element regressions (not proof certificates).""" |
| from __future__ import annotations |
| import numpy as np |
| from scipy.sparse import coo_matrix |
| from scipy.sparse.linalg import eigsh |
|
|
|
|
| def eigenvalues(edges, dirichlet=(), count=20, density=60): |
| vertices=sorted({e.u for e in edges}|{e.v for e in edges}) |
| ids={v:i for i,v in enumerate(vertices)}; total=len(ids) |
| entries=[] |
| for e in edges: |
| length=float(e.length) |
| ne=max(8,int(np.ceil(density*length))) |
| nodes=[ids[e.u]]+list(range(total,total+ne-1))+[ids[e.v]] |
| total+=ne-1 |
| h=length/ne |
| for a,b in zip(nodes[:-1],nodes[1:]): |
| for i,gi in enumerate((a,b)): |
| for j,gj in enumerate((a,b)): |
| entries.append((gi,gj,(1 if i==j else -1)/h,h*(2 if i==j else 1)/6)) |
| rows,cols,kdata,mdata=map(np.array,zip(*entries)) |
| K=coo_matrix((kdata,(rows.astype(int),cols.astype(int))),shape=(total,total)).tocsr() |
| M=coo_matrix((mdata,(rows.astype(int),cols.astype(int))),shape=(total,total)).tocsr() |
| excluded={ids[v] for v in dirichlet} |
| keep=[v for v in range(total) if v not in excluded] |
| K=K[keep,:][:,keep]; M=M[keep,:][:,keep] |
| if count>=len(keep): |
| raise ValueError('Too many requested eigenvalues for this mesh.') |
| vals=eigsh(K,k=count,M=M,sigma=-1e-5,which='LM',return_eigenvectors=False,tol=1e-10) |
| return np.sort(vals) |
|
|
