58 research outputs found

    Spectral Methods for Numerical Relativity

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    Multiscale Mesh Deformation Component Analysis with Attention-based Autoencoders

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    Deformation component analysis is a fundamental problem in geometry processing and shape understanding. Existing approaches mainly extract deformation components in local regions at a similar scale while deformations of real-world objects are usually distributed in a multi-scale manner. In this paper, we propose a novel method to exact multiscale deformation components automatically with a stacked attention-based autoencoder. The attention mechanism is designed to learn to softly weight multi-scale deformation components in active deformation regions, and the stacked attention-based autoencoder is learned to represent the deformation components at different scales. Quantitative and qualitative evaluations show that our method outperforms state-of-the-art methods. Furthermore, with the multiscale deformation components extracted by our method, the user can edit shapes in a coarse-to-fine fashion which facilitates effective modeling of new shapes.Comment: 15 page

    Sparse Surface Constraints for Combining Physics-based Elasticity Simulation and Correspondence-Free Object Reconstruction

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    We address the problem to infer physical material parameters and boundary conditions from the observed motion of a homogeneous deformable object via the solution of an inverse problem. Parameters are estimated from potentially unreliable real-world data sources such as sparse observations without correspondences. We introduce a novel Lagrangian-Eulerian optimization formulation, including a cost function that penalizes differences to observations during an optimization run. This formulation matches correspondence-free, sparse observations from a single-view depth sequence with a finite element simulation of deformable bodies. In conjunction with an efficient hexahedral discretization and a stable, implicit formulation of collisions, our method can be used in demanding situation to recover a variety of material parameters, ranging from Young's modulus and Poisson ratio to gravity and stiffness damping, and even external boundaries. In a number of tests using synthetic datasets and real-world measurements, we analyse the robustness of our approach and the convergence behavior of the numerical optimization scheme

    Rotation-Based Mixed Formulations for an Elasticity-Poroelasticity Interface Problem

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    In this paper we introduce a new formulation for the stationary poroelasticity equations written using the rotation vector and the total fluid-solid pressure as additional unknowns, and we also write an extension to the elasticity-poroelasticity problem. The transmission conditions are imposed naturally in the weak formulation, and the analysis of the effective governing equations is conducted by an application of Fredholm's alternative. We also propose a monolithically coupled mixed finite element method for the numerical solution of the problem. Its convergence properties are rigorously derived and subsequently confirmed by a set of computational tests that include applications to subsurface flow in reservoirs as well as to dentistry-oriented problems.Fondo Nacional de Desarrollo Científico y Tecnológico/[11160706]/FONDECYT/ChilePrograma Concurso Apoyo a Centros Científicos y Tecnológicos de Excelencia/[AFB170001]/PIA/ChileUCR::Sedes Regionales::Sede de OccidenteUCR::Vicerrectoría de Docencia::Ciencias Básicas::Facultad de Ciencias::Escuela de Matemátic

    An Implementation of Fully Convolutional Network for Surface Mesh Segmentation

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    This thesis presents an implementation of a 3-Dimensional triangular surface mesh segmentation architecture named Shape Fully Convolutional Network, which is proposed by Pengyu Wang and Yuan Gan in 2018. They designed a deep neural network that has a similar architecture as the Fully Convolutional Network, which provides a good segmentation result for 2D images, on 3D triangular surface meshes. In their implementation, 3D surface meshes are represented as graph structures to feed the network. There are three main barriers when applying the Fully Convolutional Network to graph-based data structures. • First, the pooling operation is much harder to apply to general graphs. • Second, the convolution order on a graph structure is unstable. • Third, the raw data of surface meshes cannot be directly applied to the network. To solve these problems, first, all the nodes inside the graph are re-ordered into a 1-dimensional list based on a multi-level graph coarsening algorithm, which allows the pooling operation to be applied as easily as a 1D pooling. Second, a self-defined generating layer is added before each convolution layer in the network to generate the neighbors of each node on the graph, and at the same time, sort all neighbors based on the L2 similarity to keep the convolution operation in a stable manner. Finally, three translation and rotation free low-level geometric features are pre-processed and used as input to train the network. This Shape Fully Convolution Network can effectively learn and predict triangular face-wise labels. In the end, to achieve a better result, the final labeling is optimized by the multi-label graph cut algorithm, which gives punishment to the predicted result based on the smoothness of the surface. The experiments show that the model can effectively learn and predict triangle-wise labels on surface meshes and yields good segmentation results

    Spectral Methods for Numerical Relativity

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    Version published online by Living Reviews in Relativity.International audienceEquations arising in General Relativity are usually too complicated to be solved analytically and one has to rely on numerical methods to solve sets of coupled partial differential equations. Among the possible choices, this paper focuses on a class called spectral methods where, typically, the various functions are expanded onto sets of orthogonal polynomials or functions. A theoretical introduction on spectral expansion is first given and a particular emphasis is put on the fast convergence of the spectral approximation. We present then different approaches to solve partial differential equations, first limiting ourselves to the one-dimensional case, with one or several domains. Generalization to more dimensions is then discussed. In particular, the case of time evolutions is carefully studied and the stability of such evolutions investigated. One then turns to results obtained by various groups in the field of General Relativity by means of spectral methods. First, works which do not involve explicit time-evolutions are discussed, going from rapidly rotating strange stars to the computation of binary black holes initial data. Finally, the evolutions of various systems of astrophysical interest are presented, from supernovae core collapse to binary black hole mergers

    Research and Education in Computational Science and Engineering

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    Over the past two decades the field of computational science and engineering (CSE) has penetrated both basic and applied research in academia, industry, and laboratories to advance discovery, optimize systems, support decision-makers, and educate the scientific and engineering workforce. Informed by centuries of theory and experiment, CSE performs computational experiments to answer questions that neither theory nor experiment alone is equipped to answer. CSE provides scientists and engineers of all persuasions with algorithmic inventions and software systems that transcend disciplines and scales. Carried on a wave of digital technology, CSE brings the power of parallelism to bear on troves of data. Mathematics-based advanced computing has become a prevalent means of discovery and innovation in essentially all areas of science, engineering, technology, and society; and the CSE community is at the core of this transformation. However, a combination of disruptive developments---including the architectural complexity of extreme-scale computing, the data revolution that engulfs the planet, and the specialization required to follow the applications to new frontiers---is redefining the scope and reach of the CSE endeavor. This report describes the rapid expansion of CSE and the challenges to sustaining its bold advances. The report also presents strategies and directions for CSE research and education for the next decade.Comment: Major revision, to appear in SIAM Revie

    Efficient spectral element methods for partial differential equations

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    In this thesis we applied a spectral element approximation to some elliptic partial differential equations. We demonstrated the difficulties related to the approximation of a discontinuous function in which the discontinuity is not fitted to the computational mesh. Such a situation gives rise to the Gibbs phenomenon. A h− p spectral element equivalent of the eXtended Finite Element Method (XFEM), which we termed the eXtended Spectral Element Method (XSEM) was developed. This was applied to some model problems. XSEM removes some of the oscillations caused by Gibbs phenomenon. We then explained that when approximating a discontinuous function, XSEM is able to capture the discontinuity precisely. We derive spectral element error estimates. The convergence of the approximations is studied. We have introduced several enrichment functions with the purpose of improving the approximation of discontinuous functions. In particular we have considered the twodimensional Poisson equation. Unfortunately, this implementation of XSEM was not able to recover spectral convergence. An alternative idea in which the discontinuity is constrained within a spectral element produces accurate SEM approximation. The Stokes problem was considered and solved using SEM coupled with an iterative PCG method. The zero volume condition on the pressure is satisfied identicaly using an alternative formulation of the continuity equation. Furthermore, we investigated the dependence of the accurency of the spectral element approximation on the weighting factor as well as the convergence properties of the preconditioner. An efficient and robust preconditioner is constructed for the Stokes problem. Exponential convergence was attained
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