SANM: A Symbolic Asymptotic Numerical Solver with Applications in Mesh Deformation

05/18/2021
by   Kai Jia, et al.
0

Solving nonlinear systems is an important problem. Numerical continuation methods efficiently solve certain nonlinear systems. The Asymptotic Numerical Method (ANM) is a powerful continuation method that usually converges faster than Newtonian methods. ANM explores the landscape of the function by following a parameterized solution curve approximated with a high-order power series. Although ANM has successfully solved a few graphics and engineering problems, prior to our work, applying ANM to new problems required significant effort because the standard ANM assumes quadratic functions, while manually deriving the power series expansion for nonquadratic systems is a tedious and challenging task. This paper presents a novel solver, SANM, that applies ANM to solve symbolically represented nonlinear systems. SANM solves such systems in a fully automated manner. SANM also extends ANM to support many nonquadratic operators, including intricate ones such as singular value decomposition. Furthermore, SANM generalizes ANM to support the implicit homotopy form. Moreover, SANM achieves high computing performance via optimized system design and implementation. We deploy SANM to solve forward and inverse elastic force equilibrium problems and controlled mesh deformation problems with a few constitutive models. Our results show that SANM converges faster than Newtonian solvers, requires little programming effort for new problems, and delivers comparable or better performance than a hand-coded, specialized ANM solver. While we demonstrate on mesh deformation problems, SANM is generic and potentially applicable to many tasks.

READ FULL TEXT
POST COMMENT

Comments

There are no comments yet.

Authors

page 1

01/22/2021

Implicit shock tracking for unsteady flows by the method of lines

A recently developed high-order implicit shock tracking (HOIST) framewor...
01/22/2020

An asymptotic preserving semi-implicit multiderivative solver

In this work we construct a multiderivative implicit-explicit (IMEX) sch...
12/24/2019

Implicit shock tracking using an optimization-based, r-adaptive, high-order discontinuous Galerkin method

A novel framework for resolving discontinuous solutions of conservation ...
05/01/2021

A robust, high-order implicit shock tracking method for simulation of complex, high-speed flows

High-order implicit shock tracking is a new class of numerical methods t...
06/19/2020

Mesh deformation techniques in fluid-structure interaction: robustness, accumulated distortion and computational efficiency

An important ingredient of any moving-mesh method for fluid-structure in...
09/20/2021

How to train your solver: A method of manufactured solutions for weakly-compressible SPH

The Weakly-Compressible Smoothed Particle Hydrodynamics (WCSPH) method i...
05/12/2018

Fast Symbolic 3D Registration Solution

3D registration has always been performed invoking singular value decomp...

Code Repositories

SANM

A symbolic asymptotic numerical solver


view repo
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.