arXiv subjects

Koki Sagiyama

Publications and source records attributed to Koki Sagiyama.

7 recordsLinked to original sources

arXiv2025

An abstraction for solving multi-domain problems using finite element methods

We introduce a new abstraction for the representation and solution of multi-domain problems using finite element methods. This is an advance over previous work in that it achieves a single higher-level abstraction that represents multi-domain problems in the mixed variational problem formalism. We implemented our new abstraction in UFL and Firedrake, and validated our implementations solving a quad-triangle mixed-cell-type problem, a hex-quad mixed-cell-type problem, and a fluid-structure interaction benchmark problem.

cs.MS↗

arXiv2024

An accurate SUPG-stabilized continuous Galerkin discretization for anisotropic heat flux in magnetic confinement fusion

We present a novel spatial discretization for the anisotropic heat conduction equation, aimed at improved accuracy at the high levels of anisotropy seen in a magnetized plasma, for example, for magnetic confinement fusion. The new discretization is based on a mixed formulation, introducing a form of the directional derivative along the magnetic field as an auxiliary variable and discretizing both the temperature and auxiliary fields in a continuous Galerkin (CG) space. Both the temperature and auxiliary variable equations are stabilized using the streamline upwind Petrov-Galerkin (SUPG) method, ensuring a better representation of the directional derivatives and therefore an overall more accurate solution. This approach can be seen as the CG-based version of our previous work (Wimmer, Southworth, Gregory, Tang, 2024), where we considered a mixed discontinuous Galerkin (DG) spatial discretization including DG-upwind stabilization. We prove consistency of the novel discretization, and demonstrate its improved accuracy over existing CG-based methods in test cases relevant to magnetic confinement fusion. This includes a long-run tokamak equilibrium sustainment scenario, demonstrating a 35% and 32% spurious heat loss for existing primal and mixed CG-based formulations versus 4% for our novel SUPG-stabilized discretization.

math.NA↗

arXiv2024

Efficient N-to-M Checkpointing Algorithm for Finite Element Simulations

In this work, we introduce a new algorithm for N-to-M checkpointing in finite element simulations. This new algorithm allows efficient saving/loading of functions representing physical quantities associated with the mesh representing the physical domain. Specifically, the algorithm allows for using different numbers of parallel processes for saving and loading, allowing for restarting and post-processing on the process count appropriate to the given phase of the simulation and other conditions. For demonstration, we implemented this algorithm in PETSc, the Portable, Extensible Toolkit for Scientific Computation, and added a convenient high-level interface into Firedrake, a system for solving partial differential equations using finite element methods. We evaluated our new implementation by saving and loading data involving 8.2 billion finite element degrees of freedom using 8,192 parallel processes on ARCHER2, the UK National Supercomputing Service.

cs.DC↗

arXiv2024

Absorbing boundary conditions for the Helmholtz equation using Gauss-Legendre quadrature reduced integrations

We introduce a new class of absorbing boundary conditions (ABCs) for the Helmholtz equation. The proposed ABCs are obtained by using $L$ discrete layers and the $Q_N$ Lagrange finite element in conjunction with the $N$-point Gauss-Legendre quadrature reduced integration rule in a specific formulation of perfectly matched layers. The proposed ABCs are classified by a tuple $(L,N)$, and achieve reflection error of order $O(R^{2LN})$ for some $R<1$. The new ABCs generalise the perfectly matched discrete layers proposed by Guddati and Lim [Int. J. Numer. Meth. Engng 66 (6) (2006) 949-977], including them as type $(L,1)$. An analysis of the proposed ABCs is performed motivated by the work of Ainsworth [J. Comput. Phys. 198 (1) (2004) 106-130]. The new ABCs facilitate numerical implementations of the Helmholtz problem with ABCs if $Q_N$ finite elements are used in the physical domain as well as give more insight into this field for the further advancement.

math.NA↗

arXiv2018

A numerical study of branching and stability of solutions to three-dimensional martensitic phase transformations using gradient-regularized, non-convex, finite strain elasticity

In the setting of continuum elasticity, phase transformations involving martensitic variants are modeled by a free energy density function that is non-convex in strain space. Here, we adopt an existing mathematical model in which we regularize the non-convex free energy density function by higher-order gradient terms at finite strain and derive boundary value problems via the standard variational argument applied to the corresponding total free energy, inspired by Toupin's theory of gradient elasticity. These gradient terms are to preclude existence of arbitrarily fine microstructures, while still allowing for existence of multiple solution branches corresponding to local minima of the total free energy; these are classified as metastable solution branches. The goal of this work is to solve the boundary value problem numerically in three dimensions, observe solution branches, and assess stability of each branch by numerically evaluating the second variation of the total free energy. We also study how these microstructures evolve as the length-scale parameter, the coefficient of the strain gradient terms in the free energy, approaches zero.

math.NA↗

arXiv2018

Unconditionally stable, second-order schemes for gradient-regularized, non-convex, finite-strain elasticity modeling martensitic phase transformations

In the setting of continuum elasticity martensitic phase transformations are characterized by a non-convex free energy density function that possesses multiple wells in strain space and includes higher-order gradient terms for regularization. Metastable martensitic microstructures, defined as solutions that are local minimizers of the total free energy, are of interest and are obtained as steady state solutions to the resulting transient formulation of Toupin's gradient elasticity at finite strain. This type of problem poses several numerical challenges including stiffness, the need for fine discretization to resolve microsstructures, and following solution branches. Stable and accurate time-integration schemes are essential to obtain meaningful solutions at reasonable computational cost. In this work we introduce two classes of unconditionally stable second-order time-integration schemes for gradient elasticity, each having relative advantages over the other. Numerical examples are shown highlighting these features.

math.NA↗

arXiv2016

Unconditionally stable, second-order accurate schemes for solid state phase transformations driven by mechano-chemical spinodal decomposition

We consider solid state phase transformations that are caused by free energy densities with domains of non-convexity in strain-composition space; we refer to the non-convex domains as mechano-chemical spinodals. The non-convexity with respect to composition and strain causes segregation into phases with different crystal structures. We work on an existing model that couples the classical Cahn-Hilliard model with Toupin's theory of gradient elasticity at finite strains. Both systems are represented by fourth-order, nonlinear, partial differential equations. The goal of this work is to develop unconditionally stable, second-order accurate time-integration schemes, motivated by the need to carry out large scale computations of dynamically evolving microstructures in three dimensions. We also introduce reduced formulations naturally derived from these proposed schemes for faster computations that are still second-order accurate. Although our method is developed and analyzed here for a specific class of mechano-chemical problems, one can readily apply the same method to develop unconditionally stable, second-order accurate schemes for any problems for which free energy density functions are multivariate polynomials of solution components and component gradients. Apart from an analysis and construction of methods, we present a suite of numerical results that demonstrate the schemes in action.

physics.comp-ph↗