Ansatz Spaces Research Articles

High order transition elements: The xNy-element concept, Part II: Dynamics

In this article, the construction of versatile transition elements for applications in dynamics is comprehensively discussed. Based on the transfinite mapping technique, also known as Coons–Gordon interpolation, piece-wise shape functions are derived such that an arbitrary number of elements with different ansatz spaces is coupled conformingly. This is an important prerequisite to enable the use of local mesh refinement strategies for regular quadrilateral discretizations. Due to the special construction of the shape functions, there are no hanging nodes that need to be taken care of. The focus in this contribution is on dynamics, and therefore, Lagrange polynomials are taken as the basis to construct shape functions for arbitrary transition elements. As a consequence, the Kronecker-delta and partition of unity properties are recovered. These properties are crucial to diagonalize the mass matrix using standard mass lumping techniques such as row-summing and diagonal scaling. The performance of the proposed family of elements is assessed by means of several benchmark examples, where the numerical rates of convergence are determined for both consistent and lumped mass matrix formulations.

A generalized inf–sup stable variational formulation for the wave equation

In this paper, we consider a variational formulation for the Dirichlet problem of the wave equation with zero boundary and initial conditions, where we use integration by parts in space and time. To prove unique solvability in a subspace of H1(Q) with Q being the space–time domain, the classical assumption is to consider the right–hand side f in L2(Q). Here, we analyze a generalized setting of this variational formulation, which allows us to prove unique solvability also for f being in the dual space of the test space, i.e., the solution operator is an isomorphism between the ansatz space and the dual of the test space. This new approach is based on a suitable extension of the ansatz space to include the information of the differential operator of the wave equation at the initial time t=0. These results are of utmost importance for the formulation and numerical analysis of unconditionally stable space–time finite element methods, and for the numerical analysis of boundary element methods to overcome the well–known norm gap in the analysis of boundary integral operators.

Mathematical modeling of spatio-temporal population dynamics and application to epidemic spreading

Agent based models (ABMs) are a useful tool for modeling spatio-temporal population dynamics, where many details can be included in the model description. Their computational cost though is very high and for stochastic ABMs a lot of individual simulations are required to sample quantities of interest. Especially, large numbers of agents render the sampling infeasible. Model reduction to a metapopulation model leads to a significant gain in computational efficiency, while preserving important dynamical properties. Based on a precise mathematical description of spatio-temporal ABMs, we present two different metapopulation approaches (stochastic and piecewise deterministic) and discuss the approximation steps between the different models within this framework. Especially, we show how the stochastic metapopulation model results from a Galerkin projection of the underlying ABM onto a finite-dimensional ansatz space. Finally, we utilize our modeling framework to provide a conceptual model for the spreading of COVID-19 that can be scaled to real-world scenarios.

Geometry of Kantorovich polytopes and support of optimizers for repulsive multi-marginal optimal transport on finite state spaces

We consider symmetric multi-marginal Kantorovich optimal transport problems on finite state spaces with uniform-marginal constraint. Hereby the symmetry of the problem refers to an assumption on the cost function as well as a corresponding restriction of the set of admissible trial states where the former enables the latter. Note that the symmetry of this setting forces us to pick for each of the considered marginals one and the same probability measure. The said problems consist of minimizing a linear objective function over a high-dimensional polytope, here referred to as Kantorovich polytope. The presented results are of split nature, computational and theoretical. Within the computational part only small numbers of marginals N and marginal sites ℓ are considered. This restriction allows us to computationally determine all extreme points of the Kantorovich polytope and investigate how many of them are in compliance with the in optimal transport typical Monge ansatz. Singling out the results for ℓ=3 discretization points and pairwise symmetric cost functions enables us to visually compare Kantorovich's to Monge's ansatz space for a varying number of marginals. Finally we present a necessary support-condition for optimizers which is inspired by the insights the said model problem on three sites provided. This result is not limited to the case of ℓ=3 sites and applies to symmetric pair-costs whose diagonal entries lie above a cost-specific threshold. In case N and ℓ display certain relationships the discussed condition provides an optimizer in Monge-form and implies its uniqueness as a solution of the considered Kantorovich optimal transport problem.

Adaptive BEM for elliptic PDE systems, part I: abstract framework, for weakly-singular integral equations

In the present work, we consider weakly-singular integral equations arising from linear second-order elliptic PDE systems with constant coefficients, including, e.g. linear elasticity. We introduce a general framework for optimal convergence of adaptive Galerkin BEM. We identify certain abstract conditions for the underlying meshes, the corresponding mesh-refinement strategy, and the ansatz spaces that guarantee that the weighted-residual error estimator is reliable and converges at optimal algebraic rate if used within an adaptive algorithm. These conditions are satisfied, e.g. for discontinuous piecewise polynomials on simplicial meshes as well as certain ansatz spaces used for isogeometric analysis. Technical contributions include the localization of (non-local) fractional Sobolev norms and local inverse estimates for the (non-local) boundary integral operators associated to the PDE system.

Kaskade 7 — A flexible finite element toolbox

Kaskade 7 is a finite element toolbox for the solution of stationary or transient systems of partial differential equations, aimed at supporting application-oriented research in numerical analysis and scientific computing. The library is written in C++ and is based on the Dune interface. The code is independent of spatial dimension and works with different grid managers. An important feature is the mix-and-match approach to discretizing systems of PDEs with different ansatz and test spaces for all variables.We describe the mathematical concepts behind the library as well as its structure, illustrating its use at several examples on the way.

$hp$-Multilevel Monte Carlo Methods for Uncertainty Quantification of Compressible Navier--Stokes Equations

We propose a novel $hp$-multilevel Monte Carlo method for the quantification of uncertainties in the compressible Navier-Stokes equations, using the Discontinuous Galerkin method as deterministic solver. The multilevel approach exploits hierarchies of uniformly refined meshes while simultaneously increasing the polynomial degree of the ansatz space. It allows for a very large range of resolutions in the physical space and thus an efficient decrease of the statistical error. We prove that the overall complexity of the $hp$-multilevel Monte Carlo method to compute the mean field with prescribed accuracy is, in best-case, of quadratic order with respect to the accuracy. We also propose a novel and simple approach to estimate a lower confidence bound for the optimal number of samples per level, which helps to prevent overestimating these quantities. The method is in particular designed for application on queue-based computing systems, where it is desirable to compute a large number of samples during one iteration, without overestimating the optimal number of samples. Our theoretical results are verified by numerical experiments for the two-dimensional compressible Navier-Stokes equations. In particular we consider a cavity flow problem from computational acoustics, demonstrating that the method is suitable to handle complex engineering problems.

Rotational test spaces for a fully-implicit FVM and FEM for the DNS of fluid-particle interaction

The paper presents a fully-implicit and stable finite element and finite volume scheme for the simulation of freely moving particles in a fluid. The developed method is based on the Petrov-Galerkin formulation of a vertex-centered finite volume method (PG-FVM) on unstructured grids. Appropriate extension of the ansatz and test spaces lead to a formulation comparable to a fictitious domain formulation. The purpose of this work is to introduce a new concept of numerical modeling reducing the mathematical overhead which many other methods require. It exploits the identification of the PG-FVM with a corresponding finite element bilinear form. The surface integrals of the finite volume scheme enable a natural incorporation of the interface forces purely based on the original bilinear operator for the fluid. As a result, there is no need to expand the system of equations to a saddle-point problem. Like for fictitious domain methods the extended scheme treats the particles as rigid parts of the fluid. The distinguishing feature compared to most existing fictitious domain methods is that there is no need for an additional Lagrange multiplier or other artificial external forces for the fluid-solid coupling. Consequently, only one single solve for the derived linear system for the fluid together with the particles is necessary and the proposed method does not require any fractional time stepping scheme to balance the interaction forces between fluid and particles. For the linear Stokes problem we will prove the stability of both schemes. Moreover, for the stationary case the conservation of mass and momentum is not violated by the extended scheme, i.e. conservativity is accomplished within the range of the underlying, unconstrained discretization scheme. The scheme is applicable for problems in two and three dimensions.

Equivalent Legendre polynomials: Numerical integration of discontinuous functions in the finite element methods

The efficiency of numerical integration of discontinuous functions is a challenge for many different computational methods. To overcome this problem, we introduce an efficient and simple numerical method for the integration of discontinuous functions on an arbitrary domain. In the proposed method, the discontinuous function over the domain is replaced by a continuous equivalent function using Legendre polynomials. The method is applicable to all Ansatz spaces with continuous basis functions, and it allows to use standard numerical integration methods in the entire domain. This imposed continuous function is called equivalent Legendre polynomial (ELP). Several numerical examples serve to demonstrate the efficiency and accuracy of the proposed method. Based on 2D and 3D problems of linear elastostatics, the introduced method is further evaluated in the concept of the fictitious domain finite element methods.

Linearizations of matrix polynomials in Newton bases

We discuss matrix polynomials expressed in a Newton basis, and the associated polynomial eigenvalue problems. Properties of the generalized ansatz spaces associated with such polynomials are proved directly by utilizing a novel representation of pencils in these spaces. Also, we show how the family of Fiedler pencils can be adapted to matrix polynomials expressed in a Newton basis. These new Newton–Fiedler pencils are shown to be strong linearizations, and some computational aspects related to them are discussed.

Hierarchical matrix approximation for the uncertainty quantification of potentials on random domains

Computing statistical quantities of interest of the solution of partial differential equations on random domains is an important and challenging task in engineering. We consider the computation of these quantities by the perturbation approach. Especially, we discuss how third order accurate expansions of the mean and the correlation can numerically be computed. These expansions become even fourth order accurate for certain types of boundary variations. The correction terms are given by the solution of correlation equations in the tensor product domain, which can efficiently be computed by means of H-matrices. They have recently been shown to be an efficient tool to solve correlation equations with rough data correlations, that is, with low Sobolev smoothness or small correlation length, in almost linear time. Numerical experiments in three dimensions for higher order ansatz spaces show the feasibility of the proposed algorithm. The application to a non-smooth domain is also included.

A fast sparse grid based space–time boundary element method for the nonstationary heat equation

This article presents a fast sparse grid based space–time boundary element method for the solution of the nonstationary heat equation. We make an indirect ansatz based on the thermal single layer potential which yields a first kind integral equation. This integral equation is discretized by Galerkin’s method with respect to the sparse tensor product of the spatial and temporal ansatz spaces. By employing the $$\mathcal {H}$$ -matrix and Toeplitz structure of the resulting discretized operators, we arrive at an algorithm which computes the approximate solution in a complexity that essentially corresponds to that of the spatial discretization. Nevertheless, the convergence rate is nearly the same as in case of a traditional discretization in full tensor product spaces.

Breaking the Curse of Dimension in Multi-Marginal Kantorovich Optimal Transport on Finite State Spaces

We present a new ansatz space for the general symmetric multi-marginal Kantorovich optimal transport problem on finite state spaces which reduces the number of unknowns from $\binom{N+\ell-1}{\ell-1}$ to $\ell\cdot(N+1)$, where $\ell$ is the number of marginal states and $N$ the number of marginals. The new ansatz space is a careful low-dimensional enlargement of the Monge class, which corresponds to $\ell\cdot(N-1)$ unknowns, and cures the insufficiency of the Monge ansatz; i.e., we show that the Kantorovich problem always admits a minimizer in the enlarged class, for arbitrary cost functions. Our results apply, in particular, to the discretization of multi-marginal optimal transport with Coulomb cost in three dimensions, which has received much recent interest due to its emergence as the strongly correlated limit of Hohenberg--Kohn density functional theory. In this context $N$ corresponds to the number of particles, motivating the interest in large $N$.

Geometric Subspace Updates with Applications to Online Adaptive Nonlinear Model Reduction

In many scientific applications, including model reduction and image processing, subspaces are used as ansatz spaces for the low-dimensional approximation and reconstruction of the state vectors of interest. We introduce a procedure for adapting an existing subspace based on information from the least-squares problem that underlies the approximation problem of interest such that the associated least-squares residual vanishes exactly. The method builds on a Riemmannian optimization procedure on the Grassmann manifold of low-dimensional subspaces, namely the Grassmannian Rank-One Update Subspace Estimation (GROUSE). We establish for GROUSE a closed-form expression for the residual function along the geodesic descent direction. Specific applications of subspace adaptation are discussed in the context of image processing and model reduction of nonlinear partial differential equation systems.

A fictitious domain method for the simulation of thermoelastic deformations in NC‐milling processes

SummaryThis paper presents a (higher‐order) finite element approach for the simulation of heat diffusion and thermoelastic deformations in NC‐milling processes. The inherent continuous material removal in the process of the simulation is taken into account via continuous removal‐dependent refinements of a paraxial hexahedron base‐mesh covering a given workpiece. These refinements rely on isotropic bisections of these hexahedrons along with subdivisions of the latter into tetrahedrons and pyramids in correspondence to a milling surface triangulation obtained from the application of the marching cubes algorithm. The resulting mesh is used for an element‐wise defined characteristic function for the milling‐dependent workpiece within that paraxial hexahedron base‐mesh. Using this characteristic function, a (higher‐order) fictitious domain method is used to compute the heat diffusion and thermoelastic deformations, where the corresponding ansatz spaces are defined for some hexahedron‐based refinement of the base‐mesh. Numerical experiments compared to real physical experiments exhibit the applicability of the proposed approach to predict deviations of the milled workpiece from its designed shape because of thermoelastic deformations in the process.

An efficient reformulation of a multiscale method for the eddy current problem

PurposeThis paper aims to improve the efficiency of a numerical method to treat the eddy current problem on a laminated material, where using a mesh that resolves each individual laminate would be too computationally expensive.Design/methodology/approachThe domain is modeled using a coarse mesh that treats the laminated material as a bulk with averaged properties. The fine-structured behavior is recovered by introducing micro-shape functions in the ansatz space. One such method is analyzed to find further model restrictions.FindingsBy using a special reformulation, it is possible to eliminate the additional degrees of freedom introduced by the multiscale ansatz at the cost of an additional modeling error that decreases with the laminate thickness.Originality/valueThe paper gives a computationally more efficient approximate variant to a known multiscale method.

Anisotropic hierarchic solid finite elements for the simulation of passive–active arterial wall models

A 3D anisotropic hierarchic solid finite element formulation is provided for the passive and active mechanical response of arteries. The artery is modeled as an anisotropic nearly incompressible hyperelastic tube consisting of two layers that correspond to the media and the adventitia. These layers are considered as a fiber-reinforced material consisting of two collagen fiber families that are symmetrically disposed and helically oriented around the tube’s axis. The numerical analysis is based on a 3D anisotropic hierarchic solid finite element formulation, including the possibility of varying the polynomial degree in all local directions as well as for all displacement components. For the purpose of verification, analytical solutions are provided for different benchmarks focusing on the aspects of compressible and incompressible 3D stretch, plane strain and plane stress pure shear, as well as a mono-layer anisotropic incompressible circular cylindrical artery of Holzapfel–Gasser–Ogden material under inflation and extension. The resulting solutions, including stress and strain distribution through the arterial wall, are plotted to point out the passive response and different activation levels. In addition the effect of various compressibility levels, including nearly incompressibility, is studied by numerical examples. The robustness and accuracy of the proposed method is demonstrated by comparing the results of different ansatz spaces.

A high-order finite element model for vibration analysis of cross-laminated timber assemblies

The vibration behavior of cross-laminated timber components in the low-frequency range can be predicted with high accuracy by the finite element method. However, the modeling of assembled cross-laminated timber components has been studied only scarcely. The three-dimensional p-version of the finite element method, which is characterized by hierarchic high-order shape functions, is well suited to consider coupling and support conditions. Furthermore, a small number of degrees of freedom can be obtained in case of thin-walled structures using p-elements with high aspect ratios and anisotropic ansatz spaces. In this article, a model for cross-laminated timber assemblies made of volumetric high-order finite elements is presented. Two representative types of connections are investigated, one with an elastomer between the cross-laminated timber components and the other without. The model is validated, and suitable ranges for the stiffness parameters of the finite elements which represent the junctions are identified.

A minimization principle for deformation-diffusion processes in polymeric hydrogels: Constitutive modeling and FE implementation

This paper presents a recently developed, innovative minimization principle for coupled deformation-diffusion processes applied to hydrogels and compares this new structure with the classical saddle point formulation in both variational foundation and finite element implementation. First, balance equations and boundary conditions associated with dissipative fluid transport in solids undergoing large deformation are shown to be rooted in a canonical minimization formulation. This two-field principle determines the deformation rate and the fluid flux, constitutively governed by the scalar free energy and the dissipation potential functions. It can be used to derive the well-known saddle point formulation by a Legendre transformation of the dissipation potential. Next, the variational potential is transformed to its incremental counterpart by means of a discretization in time, which offers an intuitive and unconstrained discretization within the finite element method. To this end, vectorial Raviart-Thomas shape functions are chosen for flux degrees of freedom in order to fulfill the required H(Div,B) conformity. The need for this ansatz space can be interpreted as a counterpart to the LBB condition that arises within the saddle point principle and is usually addressed by mixed element formulations. However, we are able to demonstrate equivalent or superior performance of the minimization principle in several representative boundary value problems. Phenomena specific to hydrogels like diffusion-induced large volume change with instability patterns in the presence of geometrical constraints are successfully modeled. The proposed variational framework can thus be validated and its significance complementary to classical approaches is underlined, with the inherent symmetry of the coupled problem as a key feature and consequence of the minimization formulation.

A Blocked Linear Method for Optimizing Large Parameter Sets in Variational Monte Carlo.

We present a modification to variational Monte Carlo's linear method optimization scheme that addresses a critical memory bottleneck while maintaining compatibility with both the traditional ground state variational principle and our recently introduced variational principle for excited states. For wave function ansatzes with tens of thousands of variables, our modification reduces the required memory per parallel process from tens of gigabytes to hundreds of megabytes, making the methodology a much better fit for modern supercomputer architectures in which data communication and per-process memory consumption are primary concerns. We verify the efficacy of the new optimization scheme in small molecule tests involving both the Hilbert space Jastrow antisymmetric geminal power ansatz and real space multi-Slater Jastrow expansions. Satisfied with its performance, we have added the optimizer to the QMCPACK software package, with which we test a systematically convergent, nonperturbative approach to excitation energies on the example of a Mott-insulating hydrogen ring.