Arbitrary Unstructured Meshes Research Articles

A conservative finite volume method for incompressible two-phase flows on unstructured meshes

A robust finite volume framework for solving incompressible two-phase flow problems based on an explicit dual-time stepping based artificial compressibility formulation has been developed by Parameswaran and Mandal (2019) for structured meshes. This artificial compressibility framework is extended to unstructured meshes in the present paper. In order to further enhance the applicability to real life engineering problems, the surface tension effect is also modeled here. To deal with arbitrary unstructured mesh topology, least square and Green-Gauss integral based methods are used. The efficacy of the proposed method is demonstrated using a set of scalar advection based test problems and a set of standard incompressible two-phase flow test problems involving gravitational, viscous and surface tension forces. The numerical results show improvement in accuracy compared to the results of the structured mesh formulation.

A Flux-Form Semi-Lagrangian advection scheme for tracer transport on arbitrary meshes

A variant of the flux-form semi-Lagrangian (FFSL) advection scheme is presented suitable for use on arbitrary unstructured meshes. The time-step used with the scheme is not constrained by the Courant number, resulting in model run-times that may be many times faster than Courant-constrained schemes. The scheme also has the property of mass conservation. This makes the scheme a potential candidate for offline simulation of tracers in unstructured sediment transport, biogeochemical or atmospheric chemistry models. We describe the numerics of the scheme and apply it to a number of idealised and real applications to demonstrate its utility. In these test cases we compare results from the offline FFSL scheme with those generated by the original hydrodynamic model which uses high order conservative tracer transport. We assess the FFSL scheme for accuracy, conservation and computational efficiency. Additionally, traditional semi-Lagrangian advection schemes are included in assessments for comparison. Results indicate that the FFSL scheme produces closer agreement with tracer distributions generated by the hydrodynamic model than the semi-Lagrange scheme, and mass conservation was vastly superior. The FFSL scheme was also an order of magnitude faster than the hydrodynamic model in certain cases. The FFSL scheme presented here therefore offers a viable mass-conserving, computationally-efficient option for use in offline transport models on arbitrary unstructured meshes.

Variational, stable, and self-consistent coupling of 3D electromagnetics to 1D transmission lines in the time domain

This work presents a new multiscale method for coupling the 3D Maxwell's equations to the 1D telegrapher's equations. While Maxwell's equations are appropriate for modeling complex electromagnetics in arbitrary-geometry domains, simulation cost for many applications (e.g. pulsed power) can be dramatically reduced by representing less complex transmission line regions of the domain with a 1D model. By assuming a transverse electromagnetic (TEM) ansatz for the solution in a transmission line region, we reduce the Maxwell's equations to the telegrapher's equations. We propose a self-consistent finite element formulation of the fully coupled system that uses boundary integrals to couple between the 3D and 1D domains and supports arbitrary unstructured 3D meshes. Additionally, by using a Lagrange multiplier to enforce continuity at the coupling interface, we allow for an absorbing boundary condition to also be applied to non-TEM modes on this boundary. We demonstrate that this feature reduces non-physical reflection and ringing of non-TEM modes off of the coupling boundary. By employing implicit time integration, we ensure a stable coupling, and we introduce an efficient method for solving the resulting linear systems. We demonstrate the accuracy of the new method on two verification problems, a transient O-wave in a rectilinear prism and a steady-state problem in a coaxial geometry, and show the efficiency and weak scalability of our implementation on a cold test of the Z-machine MITL and post-hole convolute.

Computational Morphogenesis: Morphologic constructions using polygonal discretizations

SummaryTo consistently coarsen arbitrary unstructured meshes, a computational morphogenesis process is built in conjunction with a numerical method of choice, such as the virtual element method with adaptive meshing. The morphogenesis procedure is performed by clustering elements based on a posteriori error estimation. Additionally, an edge straightening scheme is introduced to reduce the number of nodes and improve accuracy of solutions. The adaptive morphogenesis can be recursively conducted regardless of element type and mesh generation counting. To handle mesh modification events during the morphogenesis, a topology‐based data structure is employed, which provides adjacent information on unstructured meshes. Numerical results demonstrate that the adaptive mesh morphogenesis effectively handles mesh coarsening for arbitrarily shaped elements while capturing problematic regions such as those with sharp gradients or singularity.

Parallel Approximate Ideal Restriction Multigrid for Solving the S N Transport Equations

The computational kernel in solving the SN transport equations is the parallel sweep, which corresponds to directly inverting a block lower triangular linear system that arises in discretizations of the linear transport equation. Existing parallel sweep algorithms are fairly efficient on structured grids, but still have polynomial scaling, P1/d + M, for d dimensions, P processors, and M angles. Moreover, an efficient scalable parallel sweep algorithm for use on general unstructured meshes remains elusive. Recently, an algebraic multigrid (AMG) method based on approximate ideal restriction (AIR) was developed for nonsymmetric matrices and shown to be an effective solver for linear transport. Motivated by the superior scalability of the AMG methods (logarithmic in P) as well as the simplicity with which the AMG methods can be used in most situations, including on arbitrary unstructured meshes, this paper investigates the use of parallel AIR (pAIR) for solving the SN transport equations with source iteration in place of parallel sweeps. The results presented in this paper show that pAIR is a robust and scalable solver. Although sweeps are still shown to be much faster than pAIR on a structured mesh of a unit cube, pAIR is shown to perform similarly on both a structured and unstructured mesh, and offers a new, simple, black-box alternative to parallel transport sweeps.

Sdfibm: a signed distance field based discrete forcing immersed boundary method in OpenFOAM

In this paper we present the algorithm and implementation of an open-source immersed boundary code sdfibm, which is based on OpenFOAM v6 and written in C++. The immersed boundary method (“ibm” of the name) treats the velocity field as the volume average of fluid and solid velocities, and applies the volume-average discrete forcing to account for the fluid–solid interaction. The signed distance field (“sdf” of the name) representation of the solid shape, together with the proposed pyramid decomposition method, allow accurate calculations of the volume fraction field created by solids overlapping with an arbitrary unstructured fluid mesh. SDF removes the need of intersection test between the solid and fluid mesh, or the discretization and re-sampling of the solid shape. Users can freely combine different components (shapes, materials, and motion constraints) into new solids within the plain-text input file, and implement new shapes and motion constrains easily. sdfibm is an efficient and robust tool for exploring complex fluid–solid interactions in a fully-resolved sense, and can generate data for closure models in upscaling procedures. Program summaryProgram title:sdfibmCPC Library link to program files:http://dx.doi.org/10.17632/hcvtddjngv.1Licensing provisions: GNU General Public License 3Programming language: C++Nature of problem: Fluid–structure interaction and particle-laden multi-phase flow problems, with the structure/solid phase being rigid bodies. Examples range from classic flows passing stationary cylinder or sphere, to the sedimentation of a cluster of particles. Thanks to the fully-resolved direct numerical simulation, detail data of the flow field and particle motion are obtained as part of the solution, which can be used to analyze the dynamic process and to build coarse-grain models for industrial scale simulations.Solution method: The solution method can be divided into the fluid part and the solid part. For the fluid part, the Navier–Stokes equations are solved via the projection method. Because of the collocated arrangement of variables, a mesh face based flux field is used to enforce the divergence-free condition. The solid part follows Newtonian dynamics and is solved using linear multi-step method. The coupling of the two is via volume-average discrete forcing, which is an immersed boundary formulation suitable for rigid bodies. To find the forcing, the solid volume fraction at each fluid mesh cell is needed, which is calculated accurately on arbitrary unstructured meshes using signed distance field.

Method for Analytically Finding the Nullspace of Stiffness Matrix for Both Zeroth-Order and Higher Order Curl-Conforming Vector Bases in Unstructured Meshes

The stiffness matrix in a vector finite-element method has a nullspace whose size grows linearly with the matrix size. Finding it numerically can be computationally expensive. However, the nullspace is important in many applications from solving the low-frequency breakdown problem to developing fast solvers. In this article, an analytical method to generate the nullspace of the stiffness matrix is developed, not only for the zeroth-order but also for any higher order vector basis function in arbitrary unstructured meshes. Using the mesh information, we are able to analytically construct the nullspace without any difficulty, thus avoiding solving an eigenvalue problem for finding the nullspace. The proposed analytical method has been applied to a variety of 2-D and 3-D irregular meshes for various orders of vector basis functions. Comparisons with the reference nullspace obtained from a brute-force eigenvalue solution have validated the proposed analytical method. The proposed work has also been applied to efficiently solve the low-frequency breakdown problem in a full-wave solver and perform fast layout parasitics extraction.

Explicit Matrix-Free Time-Domain Method in Unstructured Meshes and Its Application to Stable Simulation of General Unsymmetrical Systems

The existing matrix-free time-domain (MFTD) method, though free of a matrix solution in arbitrary unstructured meshes, is not explicit in time marching, as a backward difference is utilized. The numerical system underlying the MFTD is unsymmetrical, and a traditional explicit simulation of such a system is absolutely unstable. In this article, we overcome this barrier and successfully develop a truly explicit MFTD method. In this method, a new explicit time marching scheme is created for simulating unsymmetrical systems, whose stability is theoretically proved and shown to be guaranteed. Meanwhile, the accuracy is not sacrificed; and the time step allowed by the traditional explicit method is not reduced to ensure the stability. As a result, we greatly improve the computational efficiency of the MFTD method without compromising its accuracy. In addition to the MFTD, the unsymmetrical systems are encountered in other numerical methods and analyses, such as the subgridding methods, the nonorthogonal FDTD methods, and the analysis of nonreciprocal problems. In this article, we show that the proposed new explicit method is a general method for stably simulating unsymmetrical systems. Hence, it can be utilized in other unsymmetrical methods to ensure the stability in an explicit time-domain simulation. The accuracy, efficiency, and stability of the proposed work have been demonstrated by extensive numerical experiments.

Matrix-Free Method for Transient Maxwell-Thermal Cosimulation in Arbitrary Unstructured Meshes

Existing electrical–thermal cosimulation methods require solving a system matrix when handling inhomogeneous materials and irregular geometries discretized into unstructured meshes. In this paper, a matrix-free method is developed for cosimulating full-wave Maxwell’s equations and the thermal diffusion equation in the time domain. The method is free of matrix solutions regardless of the shape of the element used for space discretization. A theoretical stability analysis is also developed for the coupled electrical and thermal analysis, which is nonlinear in nature. Numerical experiments on both unstructured tetrahedral and triangular prism element meshes have validated the accuracy and efficiency of the proposed method.

The finite cell method: Polygonal and tetrahedral cells

AbstractPartial differential equations arising in many physical problems are most commonly solved by using the finite element method (FEM). Despite being very versatile, the FEM has one crucial drawback when heterogeneous material systems are considered, i.e. it relies on geometry‐conforming discretization. This meshing process often constitutes a bottleneck in the simulation pipeline and therefore needs to be alleviated. One idea is to exploit the benefits of the fictitious domain concept. Here, Cartesian grids can be deployed to straightforwardly discretize an extended domain. In conjunction with higher order shape functions the recently introduced method is referred to as the finite cell method (FCM).The main objective of the contribution at hand is to extend the FCM to arbitrary unstructured meshes. In two‐dimensional application polygonal finite elements based on generalized barycentric coordinates are deployed, while the three‐dimensional implementation is based on tetrahedral finite elements. A further extension to pyramidal and pentahedral (wedge) elements is, however, straightforward.There are two distinct advantages of using unstructured meshes: (i) a local mesh refinement can easily be conducted and (ii) existing computational models can be re‐used for parametric studies. The latter point is important if virtual defects need to be introduced in the ideal model to study their effect on the material behaviour. These possibilities make the unstructured FCM a powerful numerical tool for the investigation of complex highly heterogeneous materials. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

LAD2D: A Computer Code for Detonation Fluid Dynamics Problems with Large Distortion and Internal Slip

This paper describes the LAD2D computer code, which numerically solves the equations of transient, multilaterals, compressible fluid dynamics. Of particular interest is the general capability to handle material interfaces, including slip, cavitations, or void closure. Also included is the capability to treat material strength and plasticity, as well as high explosive (HE). LAD2D uses a Lagrangian finite volume numerical technique. The method manages the sliding meshes and the internal meshes unifying as arbitrary polygonal meshes, and present a new changing connectivity of mesh technology based on the moving position of grid points with their topological relations. Of particular interest is the changing connectivity of mesh to handle the large deformation mesh and to close the gap during numerical simulation. The verification and validation (V&V) of LAD2D software was implemented based on the foundation of scientific software’s V&V method. Several applications of the changing connectivity of mesh are presented with comparison of calculations with experimental results to demonstrate the capability of the new method. The changing connectivity of mesh may be used on arbitrary geometries, using structured or unstructured meshes, and it leads to strong numerical simulation capability for multilaterals problems.

A new vertex-based limiting approach for nodal discontinuous Galerkin methods on arbitrary unstructured meshes

The nodal discontinuous Galerkin (DG) methods possess many good properties that make them very attractive for numerically solving the shallow water equations, but it is necessary to maintain numerical monotonicity by applying a slope-limiting approach to eliminate spurious oscillations. In this study, a new vertex-based slope limiter is developed for the nodal DG method on arbitrary unstructured meshes. This new limiting approach modifies the vertex-based limiter by a weighted reconstruction function, which satisfies the maximum principle and is totally free of tuning parameters. Three different weighting functions are proposed by modifying the stencil reconstruction methods from other limiters, and their performance and accuracy are compared through two shock flow problems and a laboratory-scale tsunami problem on triangular and quadrilateral meshes. The results indicate that the proposed limiter can eliminate the nonphysical oscillations efficiently and provide accurate and robust results on both triangular and quadrilateral unstructured meshes.

High order spectral difference lattice Boltzmann method for incompressible hydrodynamics

This work presents a lattice Boltzmann equation (LBE) based high order spectral difference method for incompressible flows. In the present method, the spectral difference (SD) method is adopted to discretize the convection and collision term of the LBE to obtain high order (≥3) accuracy. Because the SD scheme represents the solution as cell local polynomials and the solution polynomials have good tensor-product property, the present spectral difference lattice Boltzmann method (SD-LBM) can be implemented on arbitrary unstructured quadrilateral meshes for effective and efficient treatment of complex geometries. Thanks to only first oder PDEs involved in the LBE, no special techniques, such as hybridizable discontinuous Galerkin method (HDG), local discontinuous Galerkin method (LDG) and so on, are needed to discrete diffusion term, and thus, it simplifies the algorithm and implementation of the high order spectral difference method for simulating viscous flows. The proposed SD-LBM is validated with four incompressible flow benchmarks in two-dimensions: (a) the Poiseuille flow driven by a constant body force; (b) the lid-driven cavity flow without singularity at the two top corners–Burggraf flow; and (c) the unsteady Taylor–Green vortex flow; (d) the Blasius boundary-layer flow past a flat plate. Computational results are compared with analytical solutions of these cases and convergence studies of these cases are also given. The designed accuracy of the proposed SD-LBM is clearly verified.

Modification of Flux Correction method for accuracy improvement on unsteady problems

In the paper we introduce a new modification of the Flux Correction method (we name it Unsteady Flux Correction scheme) for solving convection-dominant problems on unstructured simplicial meshes. The original Flux Correction method possesses the third order of accuracy on steady solutions and the second order otherwise. Our modification does not degrade steady solutions of the Flux Correction method and, in addition, enjoys the third order on translationally-invariant meshes for unsteady problems. On arbitrary unstructured meshes for unsteady problems we show that the accuracy of Unsteady Flux Correction scheme is significantly higher than of the original (steady) Flux Correction method in terms of error values.

A discontinuous Galerkin method for a new class of Green–Naghdi equations on simplicial unstructured meshes

In this paper, we introduce a discontinuous Finite Element formulation on simplicial unstructured meshes for the study of free surface flows based on the fully nonlinear and weakly dispersive Green-Naghdi equations. Working with a new class of asymptotically equivalent equations, which have a simplified analytical structure, we consider a decoupling strategy: we approximate the solutions of the classical shallow water equations supplemented with a source term globally accounting for the non-hydrostatic effects and we show that this source term can be computed through the resolution of scalar elliptic second-order sub-problems. The assets of the proposed discrete formulation are: (i) the handling of arbitrary unstructured simplicial meshes, (ii) an arbitrary order of approximation in space, (iii) the exact preservation of the motionless steady states, (iv) the preservation of the water height positivity, (v) a simple way to enhance any numerical code based on the nonlinear shallow water equations. The resulting numerical model is validated through several benchmarks involving nonlinear wave transformations and run-up over complex topographies.

An implicit block LU-SGS finite-volume lattice-Boltzmann scheme for steady flows on arbitrary unstructured meshes

This work proposes a fully implicit lattice Boltzmann (LB) scheme based on finite-volume (FV) discretization on arbitrary unstructured meshes. The linear system derived from the finite-volume lattice Boltzmann equation (LBE) is solved by the block lower-upper (BLU) symmetric-Gauss-Seidel (SGS) algorithm. The proposed implicit FV-LB scheme is efficient and robust, and has a low-storage requirement. The effectiveness and efficiency of the proposed implicit FV-LB scheme are validated and verified by the simulations of three test cases in two dimensions: (a) the laminar Blasius flow over a flat plate with Re=105; (b) the steady viscous flow past a circular cylinder with Re=10, 20, and 40; and (c) the inviscid flow past a circular cylinder. The proposed implicit FV-LB scheme is shown to be not only effective and efficient for simulations of steady viscous flows, but also robust and efficient for simulations of inviscid flows in particular.

Matrix-Free Time-Domain Method for General Electromagnetic Analysis in 3-D Unstructured Meshes—Modified-Basis Formulation

We develop a new matrix-free time-domain method, which requires no matrix solution, in unstructured meshes for general 3-D electromagnetic analysis. The method handles arbitrary unstructured meshes with the same ease as a finite-element method. Meanwhile, it is free of matrix solutions manifested by a naturally diagonal mass matrix, just like a finite-difference time-domain method. Different from our previous formulation where traditional curl-conforming vector bases are employed, modified vector bases are developed in this paper to directly connect the unknown coefficients of the vector basis functions employed to represent E (or H) with the unknowns obtained from the curl of H (or E), without any need for transformation. The proposed method employs only a single mesh. It does not require any interpolation and projection to obtain one field unknown from the other. Its accuracy and stability are guaranteed theoretically. Numerous experiments on unstructured triangular prism and tetrahedral meshes, involving both homogeneous and inhomogeneous and lossy materials, demonstrate the generality, accuracy, stability, and computational efficiency of the proposed method. The modified higher order vector bases developed in this paper can also be used in any other method that employs higher order bases to obtain an explicit relationship between unknown fields and unknown coefficients of vector bases.

Finite Volume Lattice Boltzmann Method for Nearly Incompressible Flows on Arbitrary Unstructured Meshes

AbstractA genuine finite volume method based on the lattice Boltzmann equation (LBE) for nearly incompressible flows is developed. The proposed finite volume lattice Boltzmann method (FV-LBM) is grid-transparent,i.e., it requires no knowledge of cell topology, thus it can be implemented on arbitrary unstructured meshes for effective and efficient treatment of complex geometries. Due to the linear advection term in the LBE, it is easy to construct multi-dimensional schemes. In addition, inviscid and viscous fluxes are computed in one step in the LBE, as opposed to in two separate steps for the traditional finite-volume discretization of the Navier-Stokes equations. Because of its conservation constraints, the collision term of the kinetic equation can be treated implicitly without linearization or any other approximation, thus the computational efficiency is enhanced. The collision with multiple-relaxation-time (MRT) model is used in the LBE. The developed FV-LBM is of second-order convergence. The proposed FV-LBM is validated with three test cases in two-dimensions: (a) the Poiseuille flow driven by a constant body force; (b) the Blasius boundary layer; and (c) the steady flow past a cylinder at the Reynolds numbers Re=10, 20, and 40. The results verify the designed accuracy and efficacy of the proposed FV-LBM.

A flexible genuinely nonlinear approach for nonlinear wave propagation, breaking and run-up

In this paper we evaluate hybrid strategies for the solution of the Green–Naghdi system of equations for the simulation of fully nonlinear and weakly dispersive free surface waves. We consider a two step solution procedure composed of: a first step where the non-hydrostatic source term is recovered by inverting the elliptic coercive operator associated to the dispersive effects; a second step which involves the solution of the hyperbolic shallow water system with the source term, computed in the previous phase, which accounts for the non-hydrostatic effects. Appropriate numerical methods, that can be also generalized on arbitrary unstructured meshes, are used to discretize the two stages: the standard C0 Galerkin finite element method for the elliptic phase; either third order Finite Volume or third order stabilized Finite Element method for the hyperbolic phase. The discrete dispersion properties of the fully coupled schemes obtained are studied, showing accuracy close to or better than that of a fourth order finite difference method. The hybrid approach of locally reverting to the nonlinear shallow water equations is used to recover energy dissipation in breaking regions. To this scope we evaluate two strategies: simply neglecting the non-hydrostatic contribution in the hyperbolic phase; imposing a tighter coupling of the two phases, with a wave breaking indicator embedded in the elliptic phase to smoothly turn off the dispersive effects. The discrete models obtained are thoroughly tested on benchmarks involving wave dispersion, breaking and run-up, showing a very promising potential for the simulation of complex near shore wave physics in terms of accuracy and robustness.

An object-oriented implementation of a parallel Monte Carlo code for radiation transport

This paper describes the main features of a state-of-the-art Monte Carlo solver for radiation transport which has been implemented within COOLFluiD, a world-class open source object-oriented platform for scientific simulations. The Monte Carlo code makes use of efficient ray tracing algorithms (for 2D, axisymmetric and 3D arbitrary unstructured meshes) which are described in detail. The solver accuracy is first verified in testcases for which analytical solutions are available, then validated for a space re-entry flight experiment (i.e. FIRE II) for which comparisons against both experiments and reference numerical solutions are provided. Through the flexible design of the physical models, ray tracing and parallelization strategy (fully reusing the mesh decomposition inherited by the fluid simulator), the implementation was made efficient and reusable. Program summaryProgram title: COOLFluiD-MCCatalogue identifier: AEZG_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEZG_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: GNU General Public License, version 3No. of lines in distributed program, including test data, etc.: 1990165No. of bytes in distributed program, including test data, etc.: 149533288Distribution format: tar.gzProgramming language: C++.Computer: From desktops to large HPC distributed systems.Operating system: Mac OS X, Linux.Has the code been vectorized or parallelized?: Parallelized through MPIRAM: Depending on the problem size from a few Megabytes to several Gigabytes.Classification: 21.2.External routines: MPI, boost, PETSc, ParMETIS, cmakeNature of problem: Radiative processes play a fundamental role in countless science and engineering contexts, including combustion, astrophysics, atmospheric space re-entry, experiments in plasma facilities (e.g. shock tubes, arc jets). The problem we are interested in is the computation of radiative heat transfer on arbitrarily complex geometries, in particular to characterize thermal loads acting on the surface of space vehicles.Solution method: Our C++ code implements a flexible and efficient Monte Carlo algorithm making use of state-of-the-art ray tracing techniques designed for and validated on arbitrary unstructured meshes. The code is organized as a collection of dynamically linked libraries connected to the computational kernel of the open source COOLFluiD platform. The Monte Carlo method is parallelized through mesh decomposition, while reusing the same partitioned mesh (and associated data structures) on which fluid dynamics equations are solved.Unusual features: The radiation transfer code of COOLFluiD-MC offers a unique combination of ray tracing algorithms suitable for handling 2D, 2D axisymmetric and 3D simulations on arbitrary unstructured grids. The parallelization strategy is scalable and based on domain decomposition. The overall object-oriented design allows for easy integration of new models and algorithms which can be plugged in dynamically through self-registration techniques.Additional comments: The code is part of a much larger set of COOLFluiD libraries, which are fully available on Github at https://github.com/andrealani/COOLFluiD and include multiple numerical solvers, physico-chemical models, processing algorithms and interfaces to third party scientific software. The CPC distribution also includes a state-of-the-art fully implicit Finite Volume solver for Euler and Navier–Stokes systems which is used to produce the flow solution for the radiative transport algorithm described in the paper. The aerothermodynamic solver, which is used for the final testcase (FIREII) in our paper, is based on the same core Finite Volume modules but on different physico-chemical and radiation models. The latter cannot be made available through the CPC library since they rely on third party libraries (MUTATION version 2.0 and PARADE version 3.1) which cannot be distributed due to NDA.!!!!! The distribution file for this program is over 149 Mbytes and therefore is not delivered directly when download or Email is requested. Instead a html file giving details of how the program can be obtained is sent. !!!!!Running time: Depending on the size of the computational grid, if it is 2D or 3D, on the number of processors used, the running time can vary from a few seconds up to hours or days. The testcases that are included with this version of the code, all using relatively small 2D or 3D meshes, run in 1–10 s each on 8 CPU-cores. The user is, however, totally free to run the same cases on a different number of cores (more or less, according to his/her convenience).