Stable Penalty Research Articles

A bound preserving cut discontinuous Galerkin method for one dimensional hyperbolic conservation laws

In this paper we present a family of high order cut finite element methods with bound preserving properties for hyperbolic conservation laws in one space dimension. The methods are based on the discontinuous Galerkin framework and use a regular background mesh, where interior boundaries are allowed to cut through the mesh arbitrarily. Our methods include ghost penalty stabilization to handle small cut elements and a new reconstruction of the approximation on macro-elements, which are local patches consisting of cut and un-cut neighboring elements that are connected by stabilization. We show that the reconstructed solution retains conservation and order of convergence. Our lowest order scheme results in a piecewise constant solution that satisfies a maximum principle for scalar hyperbolic conservation laws. When the lowest order scheme is applied to the Euler equations, the scheme is positivity preserving in the sense that positivity of pressure and density are retained. For the high order schemes, suitable bound preserving limiters are applied to the reconstructed solution on macro-elements. In the scalar case, a maximum principle limiter is applied, which ensures that the limited approximation satisfies the maximum principle. Correspondingly, we use a positivity preserving limiter for the Euler equations, and show that our scheme is positivity preserving. In the presence of shocks additional limiting is needed to avoid oscillations, hence we apply a standard TVB limiter to the reconstructed solution. The time step restrictions are of the same order as for the corresponding discontinuous Galerkin methods on the background mesh. Numerical computations illustrate accuracy, bound preservation, and shock capturing capabilities of the proposed schemes.

Pressure and convection robust bounds for continuous interior penalty divergence-free finite element methods for the incompressible Navier–Stokes equations

Abstract In this paper, we analyze a pressure-robust method based on divergence-free mixed finite element methods with continuous interior penalty stabilization. The main goal is to prove an $O(h^{k+1/2})$ error estimate for the $L^2$ norm of the velocity in the convection dominated regime. This bound is pressure robust (the error bound of the velocity does not depend on the pressure) and also convection robust (the constants in the error bounds are independent of the Reynolds number).

A Nitsche‐based cut finite element solver for two‐phase Stefan problems

AbstractWe present a cut finite element method based on ghost penalty stabilization technique for solving two‐phase Stefan problems. The essential interfacial constraints are weakly enforced using the symmetric variant of Nitsche's method. To track the interface efficiently, the level‐set technique is employed and the front location is updated at each time step by solving a transport equation. Because this is a convection problem, we utilize the continuous interior penalty method to stabilize the finite element formulation. According to the Stefan condition, the normal velocity of the interface is proportional to the jump in the interfacial flux. To accurately approximate this quantity, we use a recent post‐processing technique that combines a ghost penalty regularization and the domain integral method on cut elements. We validate our algorithm with several numerical examples that demonstrate optimal convergence.

Upward scalable vehicle routing problem of automobile inbound logistics with pickup flexibility

Motivated by the inbound logistics of a famous automobile manufacturing company, we introduce the upward scalable vehicle routing problem (for order pickup) with time windows (USVRPTW), where the pickup from each supplier can be adjusted upward by a certain degree (pickup flexibility) based on the order volume, thus increasing the vehicle utilization and reducing logistics cost. We solve the USVRPTW exactly by a branch-and-price algorithm, where the flexibility affects the pricing problem, leading to the elementary shortest path problem having to consider resource allocation except the resource constraints. The consideration of resource allocation adds many new properties to the shortest path problem, based on which we design a tree search algorithm. We develop a heuristic algorithm based on the bipartite graph to generate initial columns for the column generation (CG) process. The algorithm can also be adopted as an efficient method for solving large-scale problems due to its ability to find near-optimal solutions quickly. We also propose the penalty stabilization method and the drill-down strategy to accelerate CG. Numerical experiments show that our designed branch-and-price algorithm outperforms the commercial solver Gurobi. The efficiency of the tree search algorithm, the heuristic algorithm, and the CG acceleration methods is also verified. Real-data experiments illustrate that the low increase in driving cost can significantly improve vehicle utilization, proving the significance of flexibility. We then provide management insights to reveal that adopting the proposed flexibility mechanism can reduce logistics cost.

Jump penalty stabilization techniques for under-resolved turbulence in discontinuous Galerkin schemes

Jump penalty stabilization techniques for under-resolved turbulence have been recently proposed for continuous and discontinuous high order Galerkin schemes [1–3]. The stabilization relies on the gradient or solution discontinuity at element interfaces to incorporate localised numerical diffusion in the numerical scheme. This diffusion acts as an implicit subgrid model and stabilizes under-resolved turbulent simulations.This paper investigates the effect of jump penalty stabilization methods (penalising gradient or solution) for stabilization and improvement of high-order discontinuous Galerkin schemes in turbulent regime. We analyze these schemes using an eigensolution analysis, a 1D non-linear Burgers equation (mimicking a turbulent cascade) and 3D turbulent Navier-Stokes simulations (Taylor-Green Vortex and Kelvin-Helmholtz Instability problems).We show that the two jump penalty stabilization techniques can stabilize under-resolved simulations thanks to the improved dispersion-dissipation characteristics (when compared to non-penalized schemes) and provide accurate results for turbulent flows. The numerical results indicate that the proposed jump penalty methods stabilize under-resolved simulations and improve the simulations, when compared to the original unpenalized scheme and to classic explicit subgrid models (Smagorinsky and Vreman).

Continuous interior penalty stabilization for divergence-free finite element methods

Abstract In this paper, we propose, analyze and test numerically a pressure-robust stabilized finite element for a linearized problem in incompressible fluid mechanics, namely, the steady Oseen equation with low viscosity. Stabilization terms are defined by jumps of different combinations of derivatives for the convective term over the element faces of the triangulation of the domain. With the help of these stabilizing terms, and the fact the finite element space is assumed to provide a point-wise divergence-free velocity, an $\mathcal O\big(h^{k+\frac 12}\big)$ error estimate in the $L^2$-norm is proved for the method (in the convection-dominated regime), and optimal order estimates in the remaining norms of the error. Numerical results supporting the theoretical findings are provided.

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 cutFEM divergence–free discretization for the stokes problem

We construct and analyze a CutFEM discretization for the Stokes problem based on the Scott–Vogelius pair. The discrete piecewise polynomial spaces are defined on macro-element triangulations which are not fitted to the smooth physical domain. Boundary conditions are imposed via penalization through the help of a Nitsche-type discretization, whereas stability with respect to small and anisotropic cuts of the bulk elements is ensured by adding local ghost penalty stabilization terms. We show stability of the scheme as well as a divergence–free property of the discrete velocity outside an O(h) neighborhood of the boundary. To mitigate the error caused by the violation of the divergence–free condition, we introduce local grad–div stabilization. The error analysis shows that the grad–div parameter can scale like O(h−1), allowing a rather heavy penalty for the violation of mass conservation, while still ensuring optimal order error estimates.

Analysis of optimal preconditioners for CutFEM

AbstractIn this article, we consider a class of unfitted finite element methods for scalar elliptic problems. These so‐called CutFEM methods use standard finite element spaces on a fixed unfitted triangulation combined with the Nitsche technique and a ghost penalty stabilization. As a model problem we consider the application of such a method to the Poisson interface problem. We introduce and analyze a new class of preconditioners that is based on a subspace decomposition approach. The unfitted finite element space is split into two subspaces, where one subspace is the standard finite element space associated to the background mesh and the second subspace is spanned by all cut basis functions corresponding to nodes on the cut elements. We will show that this splitting is stable, uniformly in the discretization parameter and in the location of the interface in the triangulation. Based on this we introduce an efficient preconditioner that is uniformly spectrally equivalent to the stiffness matrix. Using a similar splitting, it is shown that the same preconditioning approach can also be applied to a fictitious domain CutFEM discretization of the Poisson equation. Results of numerical experiments are included that illustrate optimality of such preconditioners for the Poisson interface problem and the Poisson fictitious domain problem.

Weighted Error Estimates for Transient Transport Problems Discretized Using Continuous Finite Elements with Interior Penalty Stabilization on the Gradient Jumps

In this paper we consider the semi-discretization in space of a first order scalar transport equation. For the space discretization we use standard continuous finite elements with a stabilization consisting of a penalty on the jump of the gradient over element faces. We recall some global error estimates for smooth and rough solutions and then prove a new local error estimate for the transient linear transport equation. In particular we show that for the stabilized method the effect of non-smooth features in the solution decay exponentially from the space time zone where the solution is rough so that smooth features will be transported unperturbed. Locally the L2-norm error converges with the expected order O(h^{k+frac 12}), if the exact solution is locally smooth. We then illustrate the results numerically. In particular we show the good local accuracy in the smooth zone of the stabilized method and that the standard Galerkin fails to approximate a solution that is smooth at the final time if underresolved features have been present in the solution at some time during the evolution.

Spectral Analysis of Continuous FEM for Hyperbolic PDEs: Influence of Approximation, Stabilization, and Time-Stepping

We study continuous finite element dicretizations for one dimensional hyperbolic partial differential equations. The main contribution of the paper is to provide a fully discrete spectral analysis, which is used to suggest optimal values of the CFL number and of the stabilization parameters involved in different types of stabilization operators. In particular, we analyze the streamline-upwind Petrov-Galerkin (SUPG) stabilization technique, the continuous interior penalty (CIP) stabilization method and the local projection stabilization (LPS). Three different choices for the continuous finite element space are compared: Bernstein polynomials, Lagrangian polynomials on equispaced nodes, and Lagrangian polynomials on Gauss-Lobatto cubature nodes. For the last choice, we only consider inexact quadrature based on the formulas corresponding to the degrees of freedom of the element, which allows to obtain a fully diagonal mass matrix. We also compare different time stepping strategies, namely Runge-Kutta (RK), strong stability preserving RK (SSPRK) and deferred correction time integration methods. The latter allows to alleviate the computational cost as the mass matrix inversion is replaced by the high order correction iterations. To understand the effects of these choices, both time-continuous and fully discrete Fourier analysis are performed. These allow to compare all the different combinations in terms of accuracy and stability, as well as to provide suggestions for optimal values discretization parameters involved. The results are thoroughly verified numerically both on linear and non-linear problems, and error-CPU time curves are provided. Our final conclusions suggest that cubature elements combined with SSPRK and CIP or LPS stabilization are the most promising combinations.

Spectral extended finite element method for band structure calculations in phononic crystals

This talk introduces the extended finite element method (X-FEM) on structured higher-order (spectral) finite element meshes for computing the band structure of one- and two-dimensional phononic composites. The X-FEM enables the modeling of holes and inclusions in geometry through the framework of partition-of-unity enrichment. This reduces the burden on mesh generation, since meshes do not need to conform to geometric features in the periodic domain. Further, with iterative design processes such as phononic shape optimization, the need for remeshing is eliminated. To obtain theoretical rates of convergence with higher-order extended finite elements, careful consideration of curved geometry representation and numerical integration is required. In two dimensions, we adopt rational Bezier representation of conic sections or optimized cubic Hermite functions to model implicit boundaries described by level sets. Efficient computation of weak form integrals is realized through the homogeneous numerical integration scheme—a method that uses Euler’s homogeneous function theorem and Stokes’s theorem to reduce integration to the boundary of the domain. Ghost penalty stabilization is used to improve matrix conditioning on finite elements that are cut by a hole. Band structure calculations on perforated materials as well as on two-phase phononic crystals are presented that affirm the sound accuracy and optimal convergence of the method on structured, higher-order spectral finite element meshes.

High Order Cut Discontinuous Galerkin Methods for Hyperbolic Conservation Laws in One Space Dimension

In this paper, we develop a family of high order cut discontinuous Galerkin (DG) methods for hyperbolic conservation laws in one space dimension. Ghost penalty stabilization is used to stabilize the scheme for small cut elements. Analysis shows that our proposed methods have similar stability and accuracy properties as the standard DG methods on a regular mesh. We also prove that the cut DG method with piecewise constants in space is total variation diminishing. We use the strong stability preserving Runge--Kutta method for time discretization and the time step is independent on the size of cut element. Numerical examples demonstrate the cut DG methods are high order accurate for smooth problems and perform well for discontinuous problems.

Spectral extended finite element method for band structure calculations in phononic crystals

In this paper, we compute the band structure of one- and two-dimensional phononic composites using the extended finite element method (X-FEM) on structured higher-order (spectral) finite element meshes. On using partition-of-unity enrichment in finite element analysis, the X-FEM permits use of structured finite element meshes that do not conform to the geometry of holes and inclusions. This eliminates the need for remeshing in phononic shape optimization and topology optimization studies. In two dimensions, we adopt a rational Bézier representation of curved (circular) geometries, and construct suitable material enrichment functions to model two-phase composites. A Bloch-formulation of the elastodynamic phononic eigenproblem is adopted. Efficient computation of weak form integrals with polynomial integrands is realized via the homogeneous numerical integration scheme—a method that uses Euler's homogeneous function theorem and Stokes's theorem to reduce integration to the boundary of the domain. Ghost penalty stabilization is used on finite elements that are cut by a hole. Band structure calculations on perforated (circular holes, elliptical holes, and holes defined as a level set) materials as well as on two-phase phononic crystals are presented that affirm the sound accuracy and optimal convergence of the method on structured, higher-order spectral finite element meshes. Several numerical examples are presented to demonstrate the advantages of p-refinement made possible by the spectral extended finite element method. In these examples, fourth-order spectral extended finite elements deliver O(10−8) accuracy in frequency calculations with more than thirty-fold fewer degrees-of-freedom when compared to quadratic finite elements.

A discontinuous Galerkin recovery scheme with stabilization for diffusion problems

In this work, ideas previously introduced for a discontinuous Galerkin recovery method in one dimension, that involves a penalty stabilization term, are extended to an elliptic differential equation in several dimensions and different types of boundary conditions and meshes. Using standard arguments for other existing discontinuous Galerkin methods, we show results of existence and uniqueness of the solution. Also, optimal convergence rates are proved theoretically and confirmed numerically. Likewise, the numerical experiments allow us to analyze of the effect of the stabilization parameter.

A wrinkling model for pneumatic membranes and the complementarity computational framework

The paper proposes a complementarity computational framework for the wrinkling analysis of pneumatic membranes under follower loads. Geometric and material nonlinearities are separated by using the co-rotational finite element method. A reasonable wrinkling model is proposed based on the constitutive law of bi-modulus materials in the principal stress space. To improve the convergence, a linear complementarity framework is constructed in the local frame and embedded into the global Newton–Raphson iteration. The proposed method requires no extra solution techniques to ensure convergence, compared with other solution strategies, such as a pseudo-dynamic method and a penalty stabilization method. Three benchmark tests are employed to verify the proposed model and method. The numerical results have a good agreement with the existing numerical and experimental data. Importantly, the proposed computational method can even make a more accurate prediction on the displacement response and wrinkling regions than the post-buckling analysis of thin shells in some situation.

Modeling curved interfaces without element‐partitioning in the extended finite element method

SummaryIn this paper, we model holes and material interfaces (weak discontinuities) in two‐dimensional linear elastic continua using the extended finite element method on higher‐order (spectral) finite element meshes. Arbitrary parametric curves such as rational Bézier curves and cubic Hermite curves are adopted in conjunction with the level set method to represent curved interfaces. Efficient computation of weak form integrals with polynomial integrands is realized via the homogeneous numerical integration scheme—a method that uses Euler's homogeneous function theorem and Stokes' theorem to reduce integration to the boundary of the domain. Numerical integration on cut elements requires the evaluation of a one‐dimensional integral over a parametric curve, and hence, the need to partition curved elements is eliminated. To improve stiffness matrix conditioning, ghost penalty stabilization and the Jacobi preconditioner are used. For material interface problems, we develop an enrichment function that captures weak discontinuities on spectral meshes. Taken together, we show through numerical experiments that these advances deliver optimal algebraic rates of convergence with h‐refinement (p=1,2,…,5) and exponential rates of convergence with p‐refinement (p=1,2,…,7) for elastostatic problems with holes and material inclusions on Cartesian pth‐order spectral finite element meshes.

Numerical solvers for a poromechanic problem with a moving boundary

We study a poromechanic problem in presence of a moving boundary. The poroelastic material is described by means of the Biot model while the moving boundary accounts for the effect of surface erosion of the material. We focus on the numerical approximation of the problem, in the framework of the finite element method. To avoid re-meshing along with the evolution of the boundary, we adopt the cut finite element approach. The main issue of this strategy consists of the ill-conditioning of the finite element matrices in presence of cut elements of small size. We show, by means of numerical experiments and theory, that this issue significantly decreases the performance of the numerical solver. For this reason, we propose a strategy that allows to overcome the illconditioned behavior of the discrete problem. The resulting solver is based on the fixed stress approach, used to iteratively decompose the Biot equations, combined with the ghost penalty stabilization and preconditioning applied to the pressure and displacement sub-problems respectively.

An edge‐based pressure stabilization technique for finite elements on arbitrarily anisotropic meshes

SummaryIn this paper, we analyze a stabilized equal‐order finite element approximation for the Stokes equations on anisotropic meshes. In particular, we allow arbitrary anisotropies in a subdomain, for example, along the boundary of the domain, with the only condition that a maximum angle is fulfilled in each element. This discretization is motivated by applications on moving domains as arising, for example, in fluid‐structure interaction or multiphase‐flow problems. To deal with the anisotropies, we define a modification of the original continuous interior penalty stabilization approach. We show analytically the discrete stability of the method and convergence of order in the energy norm and in the L2‐norm of the velocities. We present numerical examples for a linear Stokes problem and for a nonlinear fluid‐structure interaction problem, which substantiate the analytical results and show the capabilities of the approach.

A Nonconforming Immersed Finite Element Method for Elliptic Interface Problems

A new immersed finite element (IFE) method is developed for second-order elliptic problems with discontinuous diffusion coefficient. The IFE space is constructed based on the rotated Q1 nonconforming finite elements with the integral-value degrees of freedom. The standard nonconforming Galerkin method is employed in this IFE method without any penalty stabilization term. Error estimates in energy and L2 norms are proved to be better than $O(h\sqrt{|\log h|})$ and $O(h^2|\log h|)$, respectively, where the logarithm factors reflect jump discontinuity. Numerical results are reported to confirm our analysis.