A Geometrical Method for Low-Dimensional Representations of Simulations

Abstract

We propose a new data analysis approach for the efficient post-processing of bundles of finite element data from numerical simulations. The approach is based on the mathematical principles of symmetry. We consider the case where simulations of an industrial product are contained in the space of surface meshes embedded in R 3 . Furthermore, we assume that distance preserving transformations exist, albeit unknown, which map simulation to simulation. In this setting, a discrete Laplace-Beltrami operator can be constructed on the mesh, which is invariant to isometric transformations and therefore valid for all simulations. The eigenfunctions of such an operator are used as a common basis for all (isometric) simulations. One can use the projection coefficients instead of the full simulations for further analysis. To extend the idea of invariance, we employ a discrete Fokker-Planck operator, that in the continuous limit converges to an operator invariant to a nonlinear transformation, and use its eigendecomposition accordingly.The data analysis approach is applied to time-dependent datasets from numerical car crash simulations. One observes that only a few spectral coefficients are necessary to describe the data variability, and low dimensional structures are obtained. The eigenvectors are seen to recover different independent variation modes such as translation, rotation, or global and local deformations. An effective analysis of the data from bundles of numerical simulations is made possible, in particular an analysis for many simulations in time. *

A Compact Spectral Descriptor for Shape Deformations

“…In the engineering domain, it has been shown that spectral mesh representations using the Laplace-Beltrami operator are an adequate choice for representing several deformations in a compact way [Garcke and Iza-Teran, 2017, Iza-Teran, 2016, Iza-Teran and Garcke, 2019. Here, an approach has been proposed that uses the decomposition of one Laplace-Beltrami operator for a series of deformations, which are assumed to be isometric to a base shape.…”

“…Spectral coefficients obtained by projecting several mesh deformations in x-, yand z-directions into a common eigenbasis are used as a dimensionality reduction and visualization technique for large sets of deformed shapes resulting from crash simulations. It has also been proposed that certain eigenvectors can be interpreted as specific geometric operations on the shape, e.g., a translation in the Euclidean space [Iza-Teran and Garcke, 2019]. Furthermore, the presented approach allows to handle arbitrary discrete functions defined on the mesh, so that additional functional properties of a part, such as plastic strain or stress, may be represented in the spectral domain.…”

“…In practice, numerically -isometries will be present, which will result in only approximately the same eigenbasis. Spectral mesh decomposition still can be used accordingly since under suitable conditions the Laplacians only differ by a scaling factor in such a case [Iza-Teran and Garcke, 2019].…”

“…This approach assumes that geometries are available in a regular mesh format and that the mesh is the same, or is suitably interpolated, for all geometries in the set. A single Laplace-Beltrami operator is then computed for the set of deformations Iza-Teran, 2017, Iza-Teran andGarcke, 2019]. Observe that the operator is computed using geodesic distances along the surface of a shape, where one assumes that the deformations do not modify this distance.…”

“…, represents the relevance of that eigenvector's geometric contribution to the deformed shape's spatial representation [Iza-Teran and Garcke, 2019]. Based on this property, we propose to find a compact spectral descriptor, S, for a deformation, by selecting only those coefficients, α (•) j , that have a high relative magnitude compared to a suitable baseline.…”