Finite Element Exterior Calculus Research Articles

Canonical variables based numerical schemes for hybrid plasma models with kinetic ions and massless electrons

We study the canonical variables based numerical schemes of a hybrid model with kinetic ions and mass-less electrons. Two equivalent formulations of the hybrid model are presented with the vector potentials in different gauges and the distribution functions depending on canonical momentum (not velocity), which constitutes a pair of canonical variables with the position variable. Particle-in-cell methods are used for the distribution functions, and the vector potentials are discretized by the finite element methods in the framework of finite element exterior calculus. Splitting methods are used for the time discretizations. It is illustrated that the second formulation is numerically superior and the schemes constructed based on the anti-symmetric bracket proposed have better conservation properties and lower noise level, although the filters can be used to improve the schemes of the first formulation.

Semi-Lagrangian finite element exterior calculus for incompressible flows

We develop a semi-Lagrangian discretization of the time-dependent incompressible Navier-Stokes equations with free boundary conditions on arbitrary simplicial meshes. We recast the equations as a nonlinear transport problem for a momentum 1-form and discretize in space using methods from finite element exterior calculus. Numerical experiments show that the linearly implicit fully discrete version of the scheme enjoys excellent stability properties in the vanishing viscosity limit and is applicable to inviscid incompressible Euler flows. We obtain second-order convergence and conservation of energy is achieved through a Lagrange multiplier.

A self-consistent Hamiltonian model of the ponderomotive force and its structure preserving discretization

In the presence of an inhomogeneous oscillatory electric field, charged particles experience a net force, averaged over the oscillatory timescale, known as the ponderomotive force. We derive a one-dimensional Hamiltonian model which self-consistently couples the electromagnetic field to a plasma which experiences the ponderomotive force. We derive a family of structure preserving discretizations of the model of varying order in space and time using conforming and broken finite element exterior calculus spectral element methods. In all variants of our discretization framework, the method is found to conserve the Casimir invariants of the continuous model to machine precision and the energy to the order of the splitting method used.

Geometric Particle-In-Cell discretizations of a plasma hybrid model with kinetic ions and mass-less fluid electrons

We explore the possibilities of applying structure-preserving numerical methods to a plasma hybrid model with kinetic ions and mass-less fluid electrons satisfying the quasi-neutrality relation. The numerical schemes are derived by finite element methods in the framework of finite element exterior calculus (FEEC) for field variables, Particle-In-Cell (PIC) methods for the Vlasov equation, and splitting methods in time based on an anti-symmetric bracket proposed. Conservation properties of energy, quasi-neutrality relation, positivity of density, and divergence-free property of the magnetic field are given irrespective of the used resolution and metric. Local quasi-interpolation is used for dealing with the current terms in order to make the proposed methods more efficient. The implementation has been done in the framework of the Python package STRUPHY [1], and has been verified by extensive numerical experiments.

Analysis of a semi-implicit and structure-preserving finite element method for the incompressible MHD equations with magnetic-current formulation

In this paper, we investigate a fully discrete finite element scheme for the incompressible magnetohydrodynamic (MHD) equations with magnetic-current formulation that was introduced in Hu et al. (2016). We discretize the system by the semi-implicit Euler scheme in time and a mixed finite element approach together with finite element exterior calculus in space. The resulting scheme enjoys the structure-preserving feature that it can always produce an exactly divergence-free magnetic induction on the discrete level. The unique solvability and unconditional stability of the scheme are also proved rigorously. By utilizing the energy argument, error estimates for the velocity, magnetic induction, current density and induced electric field are further established under the low regularity hypothesis for the exact solutions. Numerical results are provided to verify the theoretical analysis and to show the effectiveness of the proposed scheme.

Data-driven Whitney forms for structure-preserving control volume analysis

Control volume analysis models physics via the exchange of generalized fluxes between subdomains. We introduce a scientific machine learning framework adopting a partition of unity architecture to identify physically-relevant control volumes, with generalized fluxes between subdomains encoded via Whitney forms. The approach provides a differentiable parameterization of geometry which may be trained in an end-to-end fashion to extract reduced models from full field data while exactly preserving physics. The architecture admits a data-driven finite element exterior calculus allowing discovery of mixed finite element spaces with closed form quadrature rules. An equivalence between Whitney forms and graph networks reveals that the geometric problem of control volume learning is equivalent to an unsupervised graph discovery problem. The framework is developed for manifolds in arbitrary dimension, with examples provided for H(div) problems in R2 establishing convergence and structure preservation properties. Finally, we consider a lithium-ion battery problem where we discover a reduced finite element space encoding transport pathways from high-fidelity microstructure resolved simulations. The approach reduces the 5.89M finite element simulation to 136 elements while reproducing pressure to under 0.1% error and preserving conservation.

A Broken FEEC Framework for Electromagnetic Problems on Mapped Multipatch Domains

We present a framework for the structure-preserving approximation of partial differential equations on mapped multipatch domains, extending the classical theory of finite element exterior calculus (FEEC) to discrete de Rham sequences which are broken, i.e., fully discontinuous across the patch interfaces. Following the Conforming/Nonconforming Galerkin (CONGA) schemes developed in Campos Pinto and Sonnendrücker (Math Comput 85:2651–2685, 2016) and Campos Pinto and Güçlü (Broken-FEEC discretizations and Hodge Laplace problems. arXiv:2109.02553, 2022), our approach is based on: (i) the identification of a conforming discrete de Rham sequence with stable commuting projection operators, (ii) the relaxation of the continuity constraints between patches, and (iii) the construction of conforming projections mapping back to the conforming subspaces, allowing to define discrete differentials on the broken sequence. This framework combines the advantages of conforming FEEC discretizations (e.g. commuting projections, discrete duality and Hodge–Helmholtz decompositions) with the data locality and implementation simplicity of interior penalty methods for discontinuous Galerkin discretizations. We apply it to several initial- and boundary-value problems, as well as eigenvalue problems arising in electromagnetics. In each case our formulations are shown to be well posed thanks to an appropriate stabilization of the jumps across the interfaces, and the solutions are extremely robust with respect to the stabilization parameter. Finally we describe a construction using tensor-product splines on mapped cartesian patches, and we detail the associated matrix operators. Our numerical experiments confirm the accuracy and stability of this discrete framework, and they allow us to verify that expected structure-preserving properties such as divergence or harmonic constraints are respected to floating-point accuracy.

Symmetry and Invariant Bases in Finite Element Exterior Calculus

AbstractWe study symmetries of bases and spanning sets in finite element exterior calculus, using representation theory. We want to know which vector-valued finite element spaces have bases invariant under permutation of vertex indices. The permutations of vertex indices correspond to the symmetry group of the simplex. That symmetry group is represented on simplicial finite element spaces by the pullback action. We determine a natural notion of invariance and sufficient conditions on the dimension and polynomial degree for the existence of invariant bases. We conjecture that these conditions are necessary too. We utilize Djoković and Malzan’s classification of monomial irreducible representations of the symmetric group and show new symmetries of the geometric decomposition and canonical isomorphisms of the finite element spaces. Explicit invariant bases with complex coefficients are constructed in dimensions two and three for different spaces of finite element differential forms.

An Arbitrary Order and Pointwise Divergence-Free Finite Element Scheme for the Incompressible 3D Navier–Stokes Equations

In this paper we introduce a new discretization of the incompressible Navier-Stokes equations. We use the Lamb identity for the advection term $(u \cdot \nabla)u$ and the general idea allows a lot of freedom in the treatment of the non-linear term. The main advantage of this scheme is that the divergence of the fluid velocity is pointwise zero at the discrete level. This exactness allows for exactly conserved quantities and pressure robustness. Discrete spaces consist of piecewise polynomials, they may be taken of arbitrary order and are already implemented in most libraries. Although the nonlinear term may be implemented as is, most proofs here are done for the linearized equation. The whole problem is expressed in the finite element exterior calculus framework. We also present numerical simulations, our codes are written with the FEniCS computing platform, version 2019.1.0.

High-Order Directional Fields

We introduce a framework for representing face-based directional fields of an arbitrary piecewise-polynomial order. Our framework is based on a primal-dual decomposition of fields, where the exact component of a field is the gradient of piecewise-polynomial conforming function, and the coexact component is defined as the adjoint of a dimensionally-consistent discrete curl operator. Our novel formulation sidesteps the difficult problem of constructing high-order non-conforming function spaces, and makes it simple to harness the flexibility of higher-order finite elements for directional-field processing. Our representation is structure-preserving, and draws on principles from finite-element exterior calculus. We demonstrate its benefits for applications such as Helmholtz-Hodge decomposition, smooth PolyVector fields, the vector heat method, and seamless parameterization.

Energy conserving particle-in-cell methods for relativistic Vlasov–Maxwell equations of laser-plasma interaction

This work studies a class of reduced relativistic Vlasov–Maxwell equations describing laser-plasma interaction. Fully discrete schemes are obtained by discretizing distribution functions using particle-in-cell methods, discretizing electromagnetic fields with compatible finite element methods in the framework of finite element exterior calculus in space, and discrete gradient methods combined with splitting methods in time. The proposed schemes are energy conserving and the discrete Poisson equations are also satisfied by the numerical solutions. Numerical experiments of parametric instability are conducted to validate the conservation properties and illustrate good long time behaviors of the numerical methods.

Dual field structure-preserving discretization of port-Hamiltonian systems using finite element exterior calculus

In this paper we propose a novel approach to discretize linear port-Hamiltonian systems while preserving the underlying structure. We present a finite element exterior calculus formulation that is able to mimetically represent conservation laws and cope with mixed open boundary conditions using a single computational mesh. The possibility of including open boundary conditions allows for modular composition of complex multi-physical systems whereas the exterior calculus formulation provides a coordinate-free treatment. Our approach relies on a dual-field representation of the physical system that is redundant at the continuous level but eliminates the need of mimicking the Hodge star operator at the discrete level. By considering the Stokes-Dirac structure representing the system together with its adjoint, which embeds the metric information directly in the codifferential, the need for an explicit discrete Hodge star is avoided altogether. By imposing the boundary conditions in a strong manner, the power balance characterizing the Stokes-Dirac structure is then retrieved at the discrete level via symplectic Runge-Kutta integrators based on Gauss-Legendre collocation points. Numerical experiments validate the convergence of the method and the conservation properties in terms of energy balance both for the wave and Maxwell equations in a three dimensional domain. For the latter example, the magnetic and electric fields preserve their divergence free nature at the discrete level.

Hybridization and postprocessing in finite element exterior calculus

We hybridize the methods of finite element exterior calculus for the Hodge–Laplace problem on differential k k -forms in R n \mathbb {R}^n . In the cases k = 0 k=0 and k = n k=n , we recover well-known primal and mixed hybrid methods for the scalar Poisson equation, while for 0 > k > n 0>k>n , we obtain new hybrid finite element methods, including methods for the vector Poisson equation in n = 2 n=2 and n = 3 n=3 dimensions. We also generalize Stenberg postprocessing [RAIRO Modél. Math. Anal. Numér. 25 (1991), pp. 151–167] from k = n k=n to arbitrary k k , proving new superconvergence estimates. Finally, we discuss how this hybridization framework may be extended to include nonconforming and hybridizable discontinuous Galerkin methods.

Computing discrete harmonic differential forms in a given cohomology class using finite element exterior calculus

Computational topology research of the past two decades has emphasized combinatorial techniques while numerical methods such as numerical linear algebra remain underutilized. While the combinatorial techniques have been very successful in diverse areas, for some applications, it is worth considering the numerical counterparts. We discuss one such application. Harmonic forms are elements of the kernel of the Hodge Laplacian operator and contain information about the topology of the manifold. If a particular cohomology class is chosen, the closed differential form with the smallest norm in that class is a harmonic form. We use these well-known facts to give an algorithm for solving the following problem: given a piecewise flat manifold simplicial complex (with or without boundary) and a closed piecewise polynomial differential form representing a cohomology class, find the discrete harmonic form in that cohomology class. We give a least squares based algorithm to solve this problem and show that the computed form satisfies the finite element exterior calculus (FEEC) equations for being a harmonic form. The piecewise polynomial spaces used are the spaces of trimmed polynomial forms, that is, arbitrary degree polynomial generalizations of Whitney forms which are used in FEEC. We also survey other methods for finding harmonic forms.

A mixed method for 3D nonlinear elasticity using finite element exterior calculus

AbstractIn this study, a mixed finite element model for 3D nonlinear elasticity using a Hu–Washizu (HW) type variational principle is presented. This mixed variational principle takes the deformed configuration and sections from its cotangent bundle as the input arguments. The critical points of the proposed HW functional enforce compatibility of these sections with the configuration, in addition to mechanical equilibrium and constitutive relations. Using this variational principle we construct a mixed FE approximation that distinguishes a vector from a 1‐form, a feature not commonly found in FE approximations for nonlinear elasticity. This distinction plays a pivotal role in identifying suitable FE spaces for approximating the 1‐forms appearing in the variational principle. These discrete approximations are constructed using ideas borrowed from finite element exterior calculus, which are in turn used to construct a discrete approximation to our HW functional. The discrete equations describing mechanical equilibrium, compatibility, and constitutive rule, are obtained by seeking extremum of the discrete functional with respect to the respective degrees of freedom. The discrete extremum problem is then solved numerically; we use Newton's method for this purpose. This mixed FE technique is then applied to a few benchmark problems wherein conventional displacement based approximations encounter locking and checker boarding. These studies help establish that our mixed FE approximation, which requires no artificial stabilizing terms, is free of these numerical bottlenecks.

A gauge-compatible Hamiltonian splitting algorithm for particle-in-cell simulations using finite element exterior calculus

A particle-in-cell algorithm is derived with a canonical Poisson structure in the formalism of finite element exterior calculus. The resulting method belongs to the class of gauge-compatible splitting algorithms, which exactly preserve gauge symmetries and their associated conservation laws via the momentum map. We numerically demonstrate this time invariance of the momentum map and its usefulness in establishing precise initial conditions with a desired initial electric field and fixed background charge. The restriction of this canonical, finite element Poisson structure to the 1X2P $1\frac {1}{2}$ -dimensional phase space is also considered and simulated numerically.

On basis constructions in finite element exterior calculus

We give a systematic self-contained exposition of how to construct geometrically decomposed bases and degrees of freedom in finite element exterior calculus. In particular, we elaborate upon a previously overlooked basis for one of the families of finite element spaces, which is of interest for implementations. Moreover, we give details for the construction of isomorphisms and duality pairings between finite element spaces. These structural results show, for example, how to transfer linear dependencies between canonical spanning sets, or how to derive the degrees of freedom.

A mixed variational principle in nonlinear elasticity using Cartan’s moving frames and implementation with finite element exterior calculus

This article is an attempt at offering a new perspective for the mechanics of solids using Cartan’s moving frames, specifically discussing a mixed variational principle in non-linear elasticity. We treat quantities defined on the co-tangent bundles of reference and deformed configurations as primary unknowns along with deformation. Such a treatment invites compatibility of the fields (defined on the co-tangent bundle) with the base-space (configurations of the solid body) so that the configuration can be realized as a subset of the Euclidean space. Using the moving frame, we rewrite the metric and connection through differential forms. These quantities are further utilized to write the deformation gradient and Cauchy–Green deformation tensor in terms of frame and co-frame fields. The geometric understanding of stress as a co-vector valued 2-form fits squarely within our overall program. We also show that, for a hyperelastic solid, an equation similar to the Doyle–Ericksen formula may be written for the co-vector part of the stress 2-form. Using this kinetic and kinematic understanding, we rewrite a mixed functional in terms of differential forms, whose extremum leads to the compatibility of deformation, constitutive relations, and equations of equilibrium. Finite element exterior calculus is then utilized to construct a finite-dimensional approximation for the differential forms appearing in the variational principle. These approximations are then used to construct a discrete functional which can be numerically extremized. The discretization leads to a mixed model as it involves independent approximations of differential forms related to stress and deformation gradient. The mixed variational principle is then specialized for the 2D case, whose discrete approximation is applied to problems in nonlinear elasticity. From the numerical study, it is found that the present discretization does not suffer from locking and related convergence issues.

Energy-preserving mixed finite element methods for the elastic wave equation

In this paper, energy-preserving mixed finite element methods corresponding to finite element exterior calculus are constructed for the first-order formulation of the elastic wave equation. The semi-discrete method conserves the system’s energies exactly. A full-discrete method employing the Crank-Nicolson method, preserves energies exactly. In addition, optimal convergence orders are obtained based on a projection-based quasi-interpolation operator. Numerical experiments confirm the theoretical results.

Trefftz co-chain calculus

We are concerned with a special class of discretizations of general linear transmission problems stated in the calculus of differential forms and posed on {mathbb {R}}^n. In the spirit of domain decomposition, we partition {mathbb {R}}^n=Omega cup Gamma cup Omega _{+}, Omega a bounded Lipschitz polyhedron, Gamma :=partial Omega , and Omega _{+} unbounded. In Omega , we employ a mesh-based discrete co-chain model for differential forms, which includes schemes like finite element exterior calculus and discrete exterior calculus. In Omega _{+}, we rely on a meshless Trefftz–Galerkin approach, i.e., we use special solutions of the homogeneous PDE as trial and test functions. Our key contribution is a unified way to couple the different discretizations across Gamma . Based on the theory of discrete Hodge operators, we derive the resulting linear system of equations. As a concrete application, we discuss an eddy-current problem in frequency domain, for which we also give numerical results.