Material-adapted refinable basis functions for elasticity simulation

Abstract

In this paper, we introduce a hierarchical construction of material-adapted refinable basis functions and associated wavelets to offer efficient coarse-graining of linear elastic objects. While spectral methods rely on global basis functions to restrict the number of degrees of freedom, our basis functions are locally supported; yet, unlike typical polynomial basis functions, they are adapted to the material inhomogeneity of the elastic object to better capture its physical properties and behavior. In particular, they share spectral approximation properties with eigenfunctions, offering a good compromise between computational complexity and accuracy. Their construction involves only linear algebra and follows a fine-to-coarse approach, leading to a block-diagonalization of the stiffness matrix where each block corresponds to an intermediate scale space of the elastic object. Once this hierarchy has been precomputed, we can simulate an object at runtime on very coarse resolution grids and still capture the correct physical behavior, with orders of magnitude speedup compared to a fine simulation. We show on a variety of heterogeneous materials that our approach outperforms all previous coarse-graining methods for elasticity.

Multiscale cholesky preconditioning for ill-conditioned problems

“…An interesting relationship between gamblets and Gaussian elimination was pointed out in Schäfer et al [2021], providing a simpler construction of gamblets via Cholesky factorization with improved asymptotic complexity. Gamblets were also used in computer animation by [Chen et al 2019] to design material-adapted, refinable basis functions and associated wavelets to offer efficient coarse-graining of elastic objects made of highly heterogeneous materials. Due to its ability to construct space-and eigenspace-localized functional spaces, this recent approach exhibits far improved homogenization properties compared to the long line of work on model reduction in animation [Nesme et al 2009;Kharevych et al 2009;Torres et al 2014;Chen et al 2017Chen et al , 2018, allowing for runtime simulations on very coarse resolution grids that still capture the correct physical behavior.…”

“…However, being focused on integral equations, they do not develop this idea further, and in particular do not provide a practical implementation. We build upon their work by relating Cholesky factorization to the construction of basis functions adapted to a given differential operator recently described in [Chen et al 2019]. As a result, we offer a much faster approach to construct the hierarchy of operator-adapted wavelets than the original algorithmic evaluation.…”

“…Our new preconditioner has its theoretical roots in a homogenization approach based on operator-adapted wavelets. From a positivedefinite stiffness matrix computed via a fine grid, the original approach of Chen et al 2019] proposes a variational construction to compute a hierarchy of refinable basis functions best adapted to a given operator L for scalar-and vectorvalued PDEs. Equipped with these resulting nested functional spaces, a Galerkin projection is performed to re-express the stiffness matrix, from which a solution is efficiently found by solving small, wellconditioned and independent linear systems corresponding to the various levels of the hierarchy.…”

“…The resulting stiffness matrix is now block-diagonal by construction, with A = diag(B −1 , ..., B 2 , B 1 , A 1 ). Details regarding sparsification through thresholding and invariance enforcement through normalization can be found in [Chen et al 2019].…”

“…While the theoretical time complexity of adapted hierarchical construction is O ( log 2 +1 ) [Owhadi and Scovel 2019], the actual implementation presented in [Chen et al 2019] involves intensive linear algebra operations throughout the sequential construction based on Eq. (4) with little opportunity for massive parallelization, thus putting a very large constant in front of the time complexity.…”