Abstract : Exact sequences of function spaces, called complexes, have played a central role in developing structure-preserving numerical methods in the framework of the finite element exterior calculus. Discrete preservation of the underlying complex structure between function spaces often preserves physical quantities and conservation laws of interest; examples include pressure-robust schemes for Navier–Stokes flows, conservation of angular momentum in linear elasticity, and propagation of the constraint equations of general relativity. In this minisymposium, we seek to investigate how far this approach can be taken by bringing together numerical analysts from across the broad spectrum of continuum mechanics problems arising in applications.
[02976] Finite element gradgrad and divdiv complexes in three dimensions
Format : Talk at Waseda University
Author(s) :
Jun Hu (Peking University)
Yizhou Liang (University of Augsburg)
Rui Ma (Beijing Institute of Technology)
Abstract : We introduce the first family of conforming discrete gradgrad complexes and divdiv complexes in three dimensions. In gradgrad complexes, the first construction of finite element spaces of $H(\operatorname{curl},\mathbb{S})$ and $H(\operatorname{div},\mathbb{T})$ was proposed, and in divdiv complexes , finite element spaces of $H(\operatorname{sym}\operatorname{curl},\mathbb{T})$ are newly constructed. We prove that these finite element complexes are exact. We also present a special family of degrees of freedom of these complexes.
[05087] Structure-preserving discretization for wave equations
Format : Talk at Waseda University
Author(s) :
Sanna Mönkölä (University of Jyväskylä)
Jonni Lohi (University of Jyväskylä)
Markus Kivioja (University of Jyväskylä)
Tytti Saksa (University of Jyväskylä)
Lauri Kettunen (University of Jyväskylä)
Tuomo Rossi (University of Jyväskylä)
Abstract : We present a comprehensive framework for linear wave equations to consider hyperbolic problems in both classical and quantum mechanics. The framework is based on differential geometry in $(d+1)$-dimensional spacetime. To discretize the equations, we use a spacetime extension of the discrete exterior calculus including a leapfrog-style evolution in the time direction. To demonstrate the efficacy of this approach, we carry out numerical simulations using a C++ software library developed at the University of Jyväskylä.
[03866] Regge finite elements and the linearized Einstein tensor
Format : Talk at Waseda University
Author(s) :
Evan Gawlik (University of Hawaii)
Michael Neunteufel (TU Wien)
Abstract : In general relativity, the linearization of the Einstein tensor plays an important role in studies of gravitational waves. We study the action of this differential operator on elements of the Regge finite element space: piecewise polynomial symmetric (0,2)-tensor fields with tangential-tangential continuity across simplex interfaces. We show that the Regge finite elements and the linearized Einstein operator fit into a commutative diagram of differential complexes that generalizes one studied by Christiansen in the lowest-order setting.
[03058] Variational structures in cochain projection based discretization of classical field theories
Format : Talk at Waseda University
Author(s) :
Brian Tran (UC San Diego)
Melvin Leok (UC San Diego)
Abstract : Compatible discretizations, such as finite element exterior calculus, provide a discretization framework that respect the cohomological structure of the de Rham complex, which can be used to systematically construct stable mixed finite element methods. Multisymplectic variational integrators are a class of geometric numerical integrators for Lagrangian and Hamiltonian field theories, and they yield methods that preserve the multisymplectic structure and momentum-conservation properties of the continuous system. In this talk, we discuss the synthesis of these two approaches, by constructing discretization of the variational principle for Lagrangian field theories utilizing structure-preserving finite element projections. In our investigation, compatible discretization by cochain projections plays a pivotal role in the preservation of the variational structure at the discrete level, allowing the discrete variational structure to essentially be the restriction of the continuum variational structure to a finite-dimensional subspace. The preservation of the variational structure at the discrete level will allow us to construct a discrete Cartan form, which encodes the variational structure of the discrete theory, and subsequently, we utilize the discrete Cartan form to naturally state discrete analogues of Noether's theorem and multisymplecticity. Time permitting, we will relate the covariant spacetime discretization to the tensor product discretization approach using variational integration in space and symplectic integration in time.
[02047] Local Exactness of de Rham Conforming Hierarchical B-spline Differential Forms
Format : Talk at Waseda University
Author(s) :
Kendrick M Shepherd (Brigham Young University)
Deepesh Toshniwal (TU Delft)
Abstract : Conservation laws present in partial differential equations arising in fluid mechanics and electromagnetics are frequently described using the de Rham sequence of differential forms. Stability of numerical methods solving these equations requires discrete preservation of these conservation laws. This talk will present sufficient local exactness criteria for a set of smooth, high-order, isogeometric, locally-refinable spline spaces in Euclidean space of arbitrary dimension in order to enable stable high-order, geometrically-precise finite element analyses.
[02065] Multigrid solvers for the de Rham complex with optimal complexity in polynomial degree
Format : Talk at Waseda University
Author(s) :
Pablo D. Brubeck (University of Oxford)
Patrick Emmet Farrell (University of Oxford)
Abstract : We present multigrid solvers for the high-order FEM de Rham complex with the same time and space complexity as sum-factorized operator application. Our approach relies on new finite elements with orthogonality properties on the reference hexahedron. The resulting sparsity enables the fast solution of the patch problems arising in the Pavarino, Arnold–Falk–Winther, and Hiptmair space decompositions, in the separable case. In the non-separable case, the method can be applied to a sparse auxiliary operator.
[04622] On partially continuous finite element spaces in variational problems of continuum mechanics
Format : Talk at Waseda University
Author(s) :
Adam Sky (University of Luxembourg)
Abstract : The use of partially continuous finite element spaces in continuum mechanics is well-established. Common applications are mixed formulations, for example in linear elasticity. While strongly symmetric finite elements for the discretisation of the stresses have been introduced, their mapping from a reference element is non-trivial. As such, in the first part of this talk we will introduce simple mappings for base functions of the Hu-Zhang element, which are applicable to hp-FEM, and discuss related problems in linear elasticity.
Linear elasticity represents one limited model of continuum mechanics. With the goal of describing more complex phenomena, generalised continua have been introduced. In the context of the linear relaxed micromorphic model, the strain field is augmented via a microdistortion field. In recent works, it was proven that the field can be defined in H(symCurl), such that well-posedness is maintained. With that in mind, the second part of this talk will discuss novel finite elements for the discretisation of the H(symCurl)-space and their application to the relaxed micromorphic sequence.
[05396] New Class of Stabilized Mixed FEM for Compressible and Incompressible Nonlinear Elasticity
Format : Talk at Waseda University
Author(s) :
Ali Gerami Matin (The George Washington University)
Abstract : In this paper, we introduce two stabilized class of mixed finite element methods with bubble functions and a perturbation method for $2$D and $3$D compressible and incompressible nonlinear elasticity. Two approaches, namely, the $L^{2}$-projection and projections of partly sobolev classes, have been employed to obtain weak formulations. Both structured and unstructured simplicial meshes have been considered to investigate effects of different meshes on our formulation. Finally, by solving some benchmark problems, we investigate performance of the stabilized elements in different circumstances. Benchmark results demonstrate that our stabilized formulation provide a robust and computationally efficient way of simulating compressible, near-incompressible, and incompressible materials. These formulations also show good performance in approximating stress and pressure in bending problems, $2$D and $3$D solids with complex geometries, and heterogeneous bodies.
[05491] Machine-Learned Whitney Forms for Structure Preservation
Format : Talk at Waseda University
Author(s) :
Jonas Albert Actor (Sandia National Laboratories)
Nathaniel Trask (Sandia National Laboratories)
Andy Huang (Sandia National Laboratories)
Abstract : For many applications, scientific machine learning techniques provide few guarantees on structure preservation or convergence rates. Towards this limitation, we introduce a partition of unity architecture to parameterize data-driven Whitney forms, allowing discovery of mixed finite element spaces. The resulting geometric parameterization extracts reduced models from full-field data while exactly preserving physics and matching expected convergence rates. We provide examples for H(div) problems in two dimensions, highlighting the capabilities of the learned Whitney form architecture.
[04438] Study of a structure preserving discretization framework for Maxwell-Klein-Gordon equations.
Format : Online Talk on Zoom
Author(s) :
Snorre Christiansen (University of Oslo)
Tore Halvorsen (University of Oslo)
Claire Scheid (Côte d'Azur University)
Abstract : We propose a numerical discretization framework for a general family of gauge invariant mechanical Lagrangian. Through the definition of a discrete gauge invariant Lagrangian, we study a fully discrete leap-frog time integration scheme based on conforming space discretizations. We prove the stability and convergence of the scheme without the a priori knowledge of the solution. We will then show how our general framework apply to the Maxwell-Klein-Gordon system.
[05131] Compatible finite elements for terrain following meshes
Format : Talk at Waseda University
Author(s) :
Karina Kowalczyk (Imperial College London)
Colin J Cotter (Imperial College London)
Abstract : In this talk we are presenting a new approach for compatible finite element discretisations for atmospheric flows on a terrain following mesh. In classical compatible finite element discretisations, the H(div)-velocity space involves the application of Piola transforms when mapping from a reference element to the physical element in order to guarantee normal continuity. In the case of a terrain following mesh, this causes an undesired coupling of the horizontal and vertical velocity components. We are proposing a new finite element space, that drops the Piola transform. For solving the equations we introduce a hybridisable formulation with trace variables supported on horizontal cell faces in order to enforce the normal continuity of the velocity in the solution. Alongside the discrete formulation for various fluid equations we discuss solver and time-stepping approaches that are compatible with them and present our latest numerical results. In the case of the Helmholtz equations we give a proof of well-posedness of the arising discrete system.
[04775] Structure-preserving discretization of momentum-based formulations of fluids using discrete exterior calculus
Format : Talk at Waseda University
Author(s) :
Christopher Eldred (Sandia National Laboratories)
Abstract : Representation of physical quantities as differential forms (using exterior calculus) has proved to be a powerful approach to formulating continuum mechanics. However, most prior work has focused on scalar-valued differential forms, and therefore electrodynamics and velocity-based formulations of fluids. This talk will present progress towards a discrete exterior calculus for (vector) bundle-valued differential forms, such as those needed to describe momentum, and illustrate its applicability for discretization of momentum-based formulations of (charged) fluid models.
