Space-time Finite Element Method Research Articles

A Space-Time Absolute Nodal Coordinate Formulation Cable Element and the Study of Its Accuracy and Efficiency

In this paper, a space-time absolute nodal coordinate formulation cable (SAC) element forming technique based on the Lagrange family of shape functions is proposed. Two distinct SAC elements, each with a distinct spatial shape function, have been generated by this method. Moreover, the external forces such as the bending moment and the air resistance formula have been accounted for. The Lagrange multiplier method, along with the concepts of replacement constraint and supplementary constraint, has been employed to provide a solution for the dynamics of constrained mechanical systems. Additionally, a constraint conversion strategy has been suggested. The solver has been constructed through Hamilton’s law of varying action. The space-time finite element method is used to solve dynamic problems, employing the Newton algorithm and quasi-Newton algorithm. The accuracy and efficiency of the solution has been verified by three simulations and one experiment. The circle-bending static simulation and the double-ended velocity impact dynamic simulation demonstrate the accuracy of the two elements. The correlation between statics and dynamics has been studied for different discretization methods and different solvers’ calculation accuracy and efficiency. Different modeling methods, time steps, order and the application of the quasi-Newton method all have a bearing on the efficiency of the solution. Finally, a comparison with an experiment in the free-pendulum simulation reveals the capability of this model to simulate dynamic problems with air resistance.

Geometrically Higher Order Unfitted Space-Time Methods for PDEs on Moving Domains

In this paper, we propose new geometrically unfitted space-time finite element methods for partial differential equations posed on moving domains of higher order accuracy in space and time. As a model problem, the convection-diffusion problem on a moving domain is studied. For geometrically higher order accuracy, we apply a parametric mapping on a background space-time tensor-product mesh. Concerning discretization in time, we consider discontinuous Galerkin, as well as related continuous (Petrov–)Galerkin and Galerkin collocation methods. For stabilization with respect to bad cut configurations and as an extension mechanism that is required for the latter two schemes, a ghost penalty stabilization is employed. The article puts an emphasis on the techniques that allow us to achieve a robust but higher order geometry handling for smooth domains. We investigate the computational properties of the respective methods in a series of numerical experiments. These include studies in different dimensions for different polynomial degrees in space and time, validating the higher order accuracy in both variables.

A Geometric Multigrid Method for Space-Time Finite Element Discretizations of the Navier–Stokes Equations and its Application to 3D Flow Simulation

We present a parallelized geometric multigrid (GMG) method, based on the cell-based Vanka smoother, for higher order space-time finite element methods (STFEM) to the incompressible Navier–Stokes equations. The STFEM is implemented as a time marching scheme. The GMG solver is applied as a preconditioner for generalized minimal residual iterations. Its performance properties are demonstrated for 2D and 3D benchmarks of flow around a cylinder. The key ingredients of the GMG approach are the construction of the local Vanka smoother over all degrees of freedom in time of the respective subinterval and its efficient application. For this, data structures that store pre-computed cell inverses of the Jacobian for all hierarchical levels and require only a reasonable amount of memory overhead are generated. The GMG method is built for the deal.II finite element library. The concepts are flexible and can be transferred to similar software platforms.

Space-Time Finite Element Method for Fully Intrinsic Equations of Geometrically Exact Beam

In this paper, a space-time finite element method based on a Galerkin-weighted residual method is proposed to solve the nonlinear fully intrinsic equations of geometrically exact beam which are first-order partial differential equations about time and space. Therefore, it is natural to discretize it in time and space simultaneously. Considering the continuity and intrinsic boundary conditions in the spatial direction and the continuity and periodic boundary conditions in the time direction, the boundary value scheme of space-time finite element for solving the full intrinsic equations is derived. This method has been successfully applied to the static analysis and dynamic response solution of the fully intrinsic equations of nonlinear geometrically exact beam. The numerical results of several examples are compared with the analytical solution, existing algorithms, and literature to illustrate the applicability, accuracy and efficiency of this method.

Spacetime finite element methods for control problems subject to the wave equation

We consider the null controllability problem for the wave equation, and analyse a stabilized finite element method formulated on a global, unstructured spacetime mesh. We prove error estimates for the approximate control given by the computational method. The proofs are based on the regularity properties of the control given by the Hilbert Uniqueness Method, together with the stability properties of the numerical scheme. Numerical experiments illustrate the results.

Adaptive space–time finite element methods for parabolic optimal control problems

Abstract We present, analyze, and test locally stabilized space–time finite element methods on fully unstructured simplicial space–time meshes for the numerical solution of space–time tracking parabolic optimal control problems with the standard L 2-regularization.We derive a priori discretization error estimates in terms of the local mesh-sizes for shape-regular meshes. The adaptive version is driven by local residual error indicators, or, alternatively, by local error indicators derived from a new functional a posteriori error estimator. The latter provides a guaranteed upper bound of the error, but is more costly than the residual error indicators. We perform numerical tests for benchmark examples having different features. In particular, we consider a discontinuous target in form of a first expanding and then contracting ball in 3d that is fixed in the 4d space– time cylinder.

A new mixed finite element formulation for reorientation in liquid crystalline elastomers

Liquid crystalline elastomers (LCE) belong to the class of soft materials and are capable of large deformations induced by temperature changes and ultraviolet irradiation. These materials are under investigation as actuator materials in light-weight structures. In order to numerically simulate LCE materials by using a space–time finite element method, a continuum model is necessary which combines a thermo-viscoelastic material formulation with a dissipative reorientation process of mesogens. Mesogens are rigid and rod-shaped molecules linked with the polymer chains of the elastomer. The dissipative reorientation of the mesogens can be described by a combination of an independent field of orientation vectors with unit length and micropolar rotational degrees of freedom. The orientation vector field follow from a partial differential equation and possesses initial–boundary conditions. The unit length of the orientation vector field for any time is guaranteed by a local rotation with the micropolar rotational degrees of freedom. The dissipative reorientation process is introduced by a local time evolution equation with respect to the micropolar rotational degrees of freedom. This is analogous to thermo-viscoelasticity and can be variationally-based formulated. In this paper, we present this new variational-based reorientation finite element formulation in a dynamical framework. We arrive at an energy–momentum consistent mixed finite element formulation with a Galerkin-based space–time integration.

Numerical Solutions of Quasilinear Parabolic Problems by a Continuous Space-Time Finite Element Scheme

In this paper, continuous space-time finite element methods are developed for approximating a class of quasilinear parabolic problems in space and in time, simultaneously. The basis of the whole approach is on a space-time variational formulation where streamline upwind terms and interface jump terms are further added for stabilizing the discretization in the direction of time. Error estimates are shown and are verified numerically through a series of numerical tests. Emphasis is placed on investigating the asymptotic convergence of the error parts, which are related to the time discretization.

Space-Time Petrov-Discontinuous Galerkin Finite Element Method for Solving Linear Convection-Diffusion Problems

The paper presents the theory of the space-time Petrov-discontinuous Galerkin finite element (PDGFE) method for the discretization of the nonstationary linear convection-diffusion problems. The PDGFE method is modified for the discontinuous Galerkin finite element (DGFE) method in the case of the symmetric interior penalty Galerkin (SIPG) scheme. PDGFE method is applied separately in space using different space gride on different time levels. We prove the properties of the bilinear form aPD, m (u, ν) (V − elliptic and continuity) stability and prove the approximate solution converges with the error of order o (h 2 + τ 3). A numerical experiment is carried out to confirm the theoretical conclusions.

NTFA-enabled goal-oriented adaptive space–time finite elements for micro-heterogeneous elastoplasticity problems

In this work, we establish a goal-oriented space–time finite element method for a class of dissipative heterogeneous materials. Those materials are modeled on both micro- and macroscale, with a scale transition of volume averaging type satisfying the Hill–Mandel condition. A nonuniform transformation field analysis is performed on the microscopic inelastic strain fields for a model reduction. Reduced variables are deduced from a space–time decomposition of those inelastic strain fields. Closed-form constitutive relations are derived from some dissipative considerations, thus resulting into a reduced order homogenization problem. The resulting model error is sufficiently small for the considered class of materials, thus leaving the discretization error of the finite element method as a main error source. For ease of error estimate, we rewrite the reduced order problem in a multifield formulation. Based on duality techniques, a backward-in-time dual problem is derived from a Lagrange method, rendering error representations of a user-defined quantity of interest. Combining a patch recovery technique, a computable error estimator is developed to quantify both spatial and temporal discretization errors. By means of a localization technique, local error estimators are used to drive a greedy adaptive refinement algorithm in space and time. The effectiveness of the resulting algorithm is confirmed by several numerical examples w.r.t. a prototype model.

High-order space–time finite element methods for the Poisson–Nernst–Planck equations: Positivity and unconditional energy stability

We present a novel class of high-order space–time finite element schemes for the Poisson–Nernst–Planck (PNP) equations. We prove that our schemes are mass conservative, positivity preserving, and unconditionally energy stable for any order of approximation. To the best of our knowledge, this is the first class of (arbitrarily) high-order accurate schemes for the PNP equations that simultaneously achieve all these three properties.This is accomplished via (1) using finite elements to directly approximate the so-called entropy variableui≔U′(ci)=log(ci) instead of the density variable ci, where U(ci)=(log(ci)−1)ci is the corresponding entropy, and (2) using a discontinuous Galerkin (DG) discretization in time. The entropy variable formulation, which was originally developed by Metti et al. (2016) under the name of a log-density formulation, guarantees both positivity of densities ci=exp(ui)>0 and a continuous-in-time energy stability result. The DG in time discretization further ensures an unconditional energy stability in the fully discrete level for any approximation order, where the lowest order case is exactly the backward Euler discretization and in this case we recover the method of Metti et al. (2016).

Enabling four-dimensional conformal hybrid meshing with cubic pyramids

The main purpose of this article is to develop a novel refinement strategy for four-dimensional hybrid meshes based on cubic pyramids. This optimal refinement strategy subdivides a given cubic pyramid into a conforming set of congruent cubic pyramids and invariant bipentatopes. The theoretical properties of the refinement strategy are rigorously analyzed and evaluated. In addition, a new class of fully symmetric quadrature rules with positive weights are generated for the cubic pyramid. These rules are capable of exactly integrating polynomials with degrees up to 12. Their effectiveness is successfully demonstrated on polynomial and transcendental functions. Broadly speaking, the refinement strategy and quadrature rules in this paper open new avenues for four-dimensional hybrid meshing, and space-time finite element methods.

Effective Viscoplastic-Softening Model Suitable for Brain Impact Modelling

In this paper, we address the numerical aspects and implementation of a nonlinear viscoplastic model of the mechanical behaviour of brain tissue to simulate the dynamic responses related to impact loads which may cause traumatic injury. Among the various viscoelastic models available, we deliberately considered modifying the Norton–Hoff model in order to introduce non-typical viscoplastic softening behaviour that imitates a brain’s response just several milliseconds after a rapid impact. We describe the discretisation and three dimensional implementation of the model, with the aim of obtaining accurate numerical results in a reasonable computational time. Due to the large scale and complexity of the problem, a parallel computation technique, using a space–time finite element method, was used to facilitate the computation boost. It is proven that, after calibrating, the introduced viscoplastic-softening model is better suited for modelling brain tissue behaviour for the specific case of rapid impact loading rather than the commonly used viscoelastic models.

A level-set based space-time finite element approach to the modelling of solidification and melting processes

We present a strategy for the numerical solution of convection-coupled phase-transition problems, with focus on solidification and melting. We solve for the temperature and flow fields over time. The position of the phase-change interface is tracked with a level-set method, which requires knowledge of the heat-flux discontinuity at the interface. In order to compute the heat-flux jump, we build upon the ghost-cell approach and extend it to the space-time finite element method. This technique does not require a local enrichment of the basis functions, such as methods like extended finite elements, and it can be easily implemented in already existing finite element codes. Verification cases for the 1D Stefan problem and the lid-driven cavity melting problem are provided. Furthermore, we show a more elaborate 2D case in view of complex applications.

The reduced-order method of continuous space-time finite element scheme for the non-stationary incompressible flows

In this paper, we mainly concern with the order reduction for the unknown solution coefficient vectors about the classical continuous space-time finite element (CCSTFE) method of the two-dimensional (2D) non-stationary incompressible Navier-Stokes equations. Therefore, we first build the CCSTFE scheme for the 2D non-stationary Navier-Stokes equations with respect to stream–vorticity functions, give the existence and stability together with convergence to the CCSTFE solutions, and rewrite the CCSTFE scheme as a matrix form. And then, we use a proper orthogonal decomposition (POD) technique to build a reduced-order extrapolated continuous space-time finite element (ROECSTFE) scheme in the matrix form including very few unknowns but possessing high enough accuracy, employ matrix analysis to demonstrate the existence and stability together with convergence of the ROECSTFE solutions such that the theoretical analysis becomes very concise and convenient, and summarize the flowchart for solving the ROECSTFE scheme. Finally, we take advantage of the numerical simulations for the back-step flow and the flow around airfoil problem to verify the correctness of theoretical results.

First failure load of sandwich beams under transient loading using a space–time coupled finite element method

We numerically analyze transient plane stress deformations of linearly elastic laminates and sandwich structures by using a coupled space–time finite element method with the weak form of governing equations derived by the least-squares method. The governing equations are written as seven first-order partial differential equations as in the state space formulation, and each variable is approximated by using piece-wise continuous basis functions. The sum of the integral over the problem (space-time) domain of squares of residuals of the governing equations, and initial and boundary conditions is minimized with respect to the nodal values of the seven variables to deduce coupled linear algebraic equations. The developed software is verified by comparing computed results for sample problems with their either analytical or numerical solutions obtained with a commercial software, ABAQUS. Using statistical method and the Tsai–Wu failure criterion, sensitivities of the maximum deflection and the first failure load to beam’s aspect ratio (AR), the facesheet-core thickness ratio (FCTR), and the facesheet-core in-plane and transverse stiffness ratios (FCISR, FCTSR) are elucidated. For sandwich beams with equal FCISR and FCTSR, the results are sensitive to the AR and the FCTR. In general, they strongly depend upon the FCTSR and its interaction with the AR and the FCTR.

Formation, Propagation and Reflection of 1D Normal Shocks in Riemann Shock Tube

The purpose of this paper is to consider 1D Riemann shock tube to investigate the formation and propagation of compression waves leading to formation, propagation and reflection of 1D normal shocks using simplified mathematical models commonly used in the published work as well as using complete mathematical models based on Conservation and Balance Laws (CBL) of classical continuum mechanics and constitutive theories for compressible viscous medium derived using entropy inequality and representation theorem. This work is aimed at resolving compression waves, the shock structure, shock formation, propagation and reflection of fully formed shocks. Evolutions obtained from the mathematical models always satisfy differentiability requirements in space and time dictated by the highest order of the derivatives of the dependent variables in the mathematical models investigated. All solutions reported in this paper including boundary conditions and initial conditions are always analytic. Solutions of the mathematical models are obtained using the space-time finite element method in which the space-time integral forms are space-time variationally consistent ensuring unconditionally stable computations during the entire evolution. Solution for a space-time strip or slab is calculated and is time marched upon convergence to obtain complete evolution for the desired space-time domain, thus ensuring time accurate evolutions. The space-time local approximation over a space-time element of a space-time strip or slab is p-version hierarchical with higher-order global differentiability in space and time, i.e., we consider scalar product approximation spaces in which k = (k1, k2) are the order of the space in space and time and p = (p1, p2) are p-levels of the approximations in space and time. Model problem studies are presented for different mathematical models and are compared with solutions obtained from the complete mathematical model based on CBL and constitutive theories for viscous compressible medium to illustrate the deficiencies and shortcomings of the simplified and approximate models in simulating correct physics of normal shocks.

Space-Time Finite Element Method for Transient and Unconfined Seepage Flow Analysis

This paper aims to develop a moving-mesh type Finite Element Method for the computation of the transient unconfined seepage flow through the porous medium. The proposed method is based on the time discontinuous Galerkin Space-Time Finite Element Method (ST/FEM). It solves the seepage problem in the saturated region. The primary unknown in ST/FEM is piezometric pressure. Fluid velocities are derived from the pressure using Darcy's law. Further, an iterative algorithm has been proposed in this paper to implement the proposed method. In each iteration step, the computation domain is updated according to the flow velocity on the phreatic boundary. Subsequently, internal nodes are moved using the mesh moving technique to accommodate the newly updated computation domain. The mesh moving technique, which is discussed in this paper, is based on an elasticity problem. ST/FEM is employed to analyze several unconfined seepage flow problems, and results of steady state solutions are compared with those available in the literature to demonstrate the efficacy of the proposed scheme. • Moving mesh type finite element method is developed for transient and unconfined seepage flow analysis. • The proposed scheme requires 2 to 3 iterations in a time step to achieve convergence. • Elasticity equation based automatic mesh moving technique is employed to move the internal nodes. • Initial computation domain significantly affects the deformation characteristics of mesh. • Proposed iterative scheme can be employed with any type of mesh moving technique.

A partitioned solver for compressible/incompressible fluid flow and light structure

In this work, a partitioned fluid-structure interaction solver is presented. Fluid flow problem is solved with time-discontinuous deforming domain stabilized space-time finite element method. Flow is computed with pressure primitive variables which permit to use the same numerical technique for both compressible and incompressible regimes. Elastic deformation of the structure is modelled in the Lagrangian frame of reference with Saint-Venant Kirchhoff and Neo-Hookean material models - both are non-linear and valid for large deformations. Structure equations are discretized with Galerkin finite element method for space and with generalized-alpha method for the time. Mesh motion is modelled with the elastic deformation method. An implicit algorithm is presented to couple the different solvers. The details are provided on the implementation of the solvers in parallel software. The numerical code is verified and validated on several compressible and incompressible flow benchmarks widely used in the literature. The results demonstrate that the developed solver successfully detects the accurate interaction between fluid and structure.

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.