An adaptive finite element method for transient compressible flows

An adaptive finite element method for the solution of time dependent strongly compressible flows in two dimensions is described. The computational domain is represented by an unstructured assembly of linear triangular elements and the mesh adaptation is achieved by local regeneration of the grid, using an error estimation procedure coupled to an automatic triangular mesh generator. Problems involving shock propagation are solved to illustrate the numerical performance of the proposed approach.     

3D NURBS‐enhanced finite element method (NEFEM)

This paper presents the extension of the recently proposed NURBS‐enhanced finite element method (NEFEM) to 3D domains. NEFEM is able to exactly represent the geometry of the computational domain by means of its CAD boundary representation with non‐uniform rational B‐splines (NURBS) surfaces. Specific strategies for interpolation and numerical integration are presented for those elements affected by the NURBS boundary representation. For elements not intersecting the boundary, a standard finite element rationale is used, preserving the efficiency of the classical FEM. In 3D NEFEM special attention must be paid to geometric issues that are easily treated in the 2D implementation. Several numerical examples show the performance and benefits of NEFEM compared with standard isoparametric or cartesian finite elements. NEFEM is a powerful strategy to efficiently treat curved boundaries and it avoids excessive mesh refinement to capture small geometric features. Copyright © 2011 John Wiley & Sons, Ltd.     

Temporal‐spatial dispersion and stability analysis of finite element method in explicit elastodynamics

This work presents the temporal‐spatial (full) dispersion and stability analysis of plane square linear and biquadratic serendipity finite elements in explicit numerical solution of transient elastodynamic problems. Here, the central difference method, as an explicit time integrator, is exploited. The paper complements and extends the previous work on spatial/grid dispersion analysis of plane square biquadratic serendipity finite elements. We report on a computational strategy for temporal‐spatial dispersion relationships, where eigenfrequencies from grid/spatial dispersion analysis are adjusted to comply with the time integration method. Besides that, an ‘optimal’ lumped mass matrix for the studied finite element types is proposed and investigated. Based on the temporal‐spatial dispersion and stability analysis, relationships suggesting the ‘proper’ choice of mesh size and time step size from knowledge of the loading spectrum are presented. Copyright © 2015 John Wiley & Sons, Ltd.     

Exploiting semantics of temporal multi‐scale methods to optimize multi‐level mesh partitioning

Multi‐scale problems are often solved by decomposing the problem domain into multiple subdomains, solving them independently using different levels of spatial and temporal refinement, and coupling the subdomain solutions back to obtain the global solution. Most commonly, finite elements are used for spatial discretization, and finite difference time stepping is used for time integration. Given a finite element mesh for the global problem domain, the number of possible decompositions into subdomains and the possible choices for associated time steps is exponentially large, and the computational costs associated with different decompositions can vary by orders of magnitude. The problem of finding an optimal decomposition and the associated time discretization that minimizes computational costs while maintaining accuracy is nontrivial. Existing mesh partitioning tools, such as METIS, overlook the constraints posed by multi‐scale methods and lead to suboptimal partitions with a high performance penalty. We present a multi‐level mesh partitioning approach that exploits domain‐specific knowledge of multi‐scale methods to produce nearly optimal mesh partitions and associated time steps automatically. Results show that for multi‐scale problems, our approach produces decompositions that outperform those produced by state‐of‐the‐art partitioners like METIS and even those that are manually constructed by domain experts. Copyright © 2017 John Wiley & Sons, Ltd.     

An optimal high‐order non‐reflecting finite element scheme for wave scattering problems

A new finite element scheme is proposed for the numerical solution of time‐harmonic wave scattering problems in unbounded domains. The infinite domain in truncated via an artificial boundary ?? which encloses a finite computational domain Ω. On ?? a local high‐order non‐reflecting boundary condition (NRBC) is applied which is constructed to be optimal in a certain sense. This NRBC is implemented in a special way, by using auxiliary variables along the boundary ??, so that it involves no high‐order derivatives regardless of its order. The order of the scheme is simply an input parameter, and it may be arbitrarily high. This leads to a symmetric finite element formulation where standard C⁰ finite elements are used in Ω. The performance of the method is demonstrated via numerical examples, and it is compared to other NRBC‐based schemes. The method is shown to be highly accurate and stable, and to lead to a well‐conditioned matrix problem. Copyright © 2002 John Wiley & Sons, Ltd.     

Lagrangian finite element treatment of transient vibration/acoustics of biosolids immersed in fluids

Superposition principle is used to separate the incident acoustic wave from the scattered and radiated waves in a displacement‐based finite element model. An absorbing boundary condition is applied to the perturbation part of the displacement. Linear constitutive equation allows for inhomogeneous, anisotropic materials, both fluids and solids. Displacement‐based finite elements are used for all materials in the computational volume. Robust performance for materials with limited compressibility is achieved using assumed‐strain nodally integrated simplex elements or incompatible‐mode brick elements. A centered‐difference time‐stepping algorithm is formulated to handle general damping accurately and efficiently. Verification problems (response of empty steel cylinder immersed in water to a step plane wave, and scattering of harmonic plane waves from an elastic sphere) are discussed for assumed‐strain simplex and for voxel‐based brick finite element models. A voxel‐based modeling scheme for complex biological geometries is described, and two illustrative results are presented from the bioacoustics application domain: reception of sound by the human ear and simulation of biosonar in beaked whales. Copyright © 2007 John Wiley & Sons, Ltd.     

An immersed finite element method with integral equation correction

We propose a robust immersed finite element method in which an integral equation formulation is used to enforce essential boundary conditions. The solution of a boundary value problem is expressed as the superposition of a finite element solution and an integral equation solution. For computing the finite element solution, the physical domain is embedded into a slightly larger Cartesian (box‐shaped) domain and is discretized using a block‐structured mesh. The defect in the essential boundary conditions, which occurs along the physical domain boundaries, is subsequently corrected with an integral equation method. In order to facilitate the mapping between the finite element and integral equation solutions, the physical domain boundary is represented with a signed distance function on the block‐structured mesh. As a result, only a boundary mesh of the physical domain is necessary and no domain mesh needs to be generated, except for the non‐boundary‐conforming block‐structured mesh. The overall approach is first presented for the Poisson equation and then generalized to incompressible viscous flow equations. As an example of fluid–structure coupling, the settling of a heavy rigid particle in a closed tank is considered. Copyright © 2010 John Wiley & Sons, Ltd.     

Numerical modelling of elastic wave scattering in frequency domain by the partition of unity finite element method

In this paper, we investigate a numerical approach based on the partition of unity finite element method, for the time‐harmonic elastic wave equations. The aim of the proposed work is to accurately model two‐dimensional elastic wave problems with fewer elements, capable of containing many wavelengths per nodal spacing, and without refining the mesh at each frequency. The approximation of the displacement field is performed via the standard finite element shape functions, enriched by superimposing pressure and shear plane wave basis, which incorporate knowledge of the wave propagation. A variational framework able to handle mixed boundary conditions is described. Numerical examples dealing with the radiation and the scattering of elastic waves by a circular body are presented. The results show the performance of the proposed method in both accuracy and efficiency. Copyright © 2008 John Wiley & Sons, Ltd.     

Hybrid time domain solvers for the Maxwell equations in 2D

We present a new approach to time domain hybrid schemes for the Maxwell equations. By combining the classical FD‐TD scheme with two unstructured solvers, one explicit finite volume solver and one implicit finite element solver, we achieve a very efficient and flexible second‐order scheme. The second‐order accuracy of the hybrid scheme is verified through convergence studies on perfectly conducting as well as dielectric and diamagnetic circular cylinders. The numerical results also show its superiority to the FD‐TD scheme. Copyright © 2002 John Wiley & Sons, Ltd.     

Converting a tetrahedral mesh to a prism–tetrahedral hybrid mesh for FEM accuracy and efficiency

This paper presents a computational method for converting a tetrahedral mesh to a prism–tetrahedral hybrid mesh for improved solution accuracy and computational efficiency of finite element analysis. The proposed method performs this conversion by inserting layers of prism elements and deleting tetrahedral elements in sweepable sub‐domains, in which cross‐sections remain topologically identical and geometrically similar along a certain sweeping path. The total number of finite elements is reduced because roughly three tetrahedral elements are converted to one prism element. The solution accuracy of the finite element analysis improves since a prism element yields a more accurate solution than a tetrahedral element due to the presence of higher‐order terms in the shape function. Only previously known method for creating such a prism–tetrahedral hybrid mesh was to manually decompose a target volume into sweepable and non‐sweepable sub‐volumes and mesh each of the sub‐volumes separately. Unlike the previous method, the proposed method starts from a cross‐section of a tetrahedral mesh and replaces the tetrahedral elements with layers of prism elements until prescribed quality criteria can no longer be satisfied. A series of computational fluid dynamics simulations and structural analyses have been conducted, and the results verified a better performance of prism–tetrahedral hybrid mesh. Copyright © 2009 John Wiley & Sons, Ltd.     

On structural shape optimization using an embedding domain discretization technique

This contribution presents a novel approach to structural shape optimization that relies on an embedding domain discretization technique. The evolving shape design is embedded within a uniform finite element background mesh which is then used for the solution of the physical state problem throughout the course of the optimization. We consider a boundary tracking procedure based on adaptive mesh refinement to separate between interior elements, exterior elements, and elements intersected by the physical domain boundary. A selective domain integration procedure is employed to account for the geometric mismatch between the uniform embedding domain discretization and the evolving structural component. Thereby, we avoid the need to provide a finite element mesh that conforms to the structural component for every design iteration, as it is the case for a standard Lagrangian approach to structural shape optimization. Still, we adopt an explicit shape parametrization that allows for a direct manipulation of boundary vertices for the design evolution process. In order to avoid irregular and impracticable design updates, we consider a geometric regularization technique to render feasible descent directions for the course of the optimization. Copyright © 2016 John Wiley & Sons, Ltd.     

Computational continua for linear elastic heterogeneous solids on unstructured finite element meshes

The computational continua framework, which is a variant of higher‐order computational homogenization theories that is free of scale separation, does not require higher‐order finite element continuity, and is free of higher‐order boundary conditions, has been generalized to unstructured meshes. The salient features of the proposed generalization are (i) a nonlocal quadrature scheme for distorted elements that accounts for unit cell distortion in the parent element domain and (ii) an approximate variant of the nonlocal quadrature that eliminates the cost of computing positions of the quadrature points in the preprocessing stage. The performance of the computational continua framework on unstructured meshes has been compared to the first‐order homogenization theory and the direct numerical simulation.     

The generation of triangular meshes for NURBS‐enhanced FEM

This paper presents the first method that enables the fully automatic generation of triangular meshes suitable for the so‐called non‐uniform rational B‐spline (NURBS)‐enhanced finite element method (NEFEM). The meshes generated with the proposed approach account for the computer‐aided design boundary representation of the domain given by NURBS curves. The characteristic element size is completely independent of the geometric complexity and of the presence of very small geometric features. The proposed strategy allows to circumvent the time‐consuming process of de‐featuring complex geometric models before a finite element mesh suitable for the analysis can be produced. A generalisation of the original definition of a NEFEM element is also proposed, enabling to treat more complicated elements with an edge defined by several NURBS curves or more than one edge defined by different NURBS. Three examples of increasing difficulty demonstrate the applicability of the proposed approach and illustrate the advantages compared with those of traditional finite element mesh generators. Finally, a simulation of an electromagnetic scattering problem is considered to show the applicability of the generated meshes for finite element analysis. ©2016 The Authors. International Journal for Numerical Methods in Engineering published by John Wiley & Sons Ltd.     

Improvement of PUFEM for the numerical solution of high‐frequency elastic wave scattering on unstructured triangular mesh grids

The purpose of this paper, which builds on previous work (Int. J. Numer. Meth. Engng 2009; 77 :1646–1669), is to improve a numerical scheme based on the partition of unity finite element method (PUFEM) for the solution of the time harmonic elastic wave equations. The approach consists to approximate the displacement field by the standard finite element shape functions, enriched locally by superimposing pressure (P) and shear (S) plane waves. The aim is to accurately model two‐dimensional elastic wave problems on relatively coarse mesh grids, capable of containing many wavelengths per nodal spacing, for wide ranges of frequencies. This allows us to relax the traditional requirement of about 10 nodal points per S wavelength. In this work, an exact integration scheme for the linear triangular finite element is developed to evaluate the oscillatory integrals arising from the use of the PUFEM. The main contribution here consists in developing an explicit closed‐form solution for two‐dimensional wave‐based integrals, when the phase variation is linear in the local coordinate element system. The evaluation of the element mass matrix is performed from appropriate edge integrals. All other element matrices, obtained by adequate splitting of the element stress tensor matrix, are simply deduced from the element mass matrix entries. The results show clearly that the proposed integration scheme evaluates accurately the entries of the global matrix with drastic reduction of the computational time. Numerical tests dealing with the scattering of S elastic plane waves by a circular rigid body show that, for the same discretization level, it is possible to improve the accuracy by using large elements associated with high numbers of approximating plane waves rather than using small elements with less plane waves. However, this increases the conditioning and the fill‐in of the global matrix. At high frequency, it is even possible to push the number of degrees of freedom per S wavelength under 2 and still achieve good accuracy. Finally, some remarks on the choice of the numbers of P and S plane waves leading to better accuracy and conditioning are discussed. Copyright © 2010 John Wiley & Sons, Ltd.     

A parallel two‐level domain decomposition based one‐shot method for shape optimization problems

A two‐level domain decomposition method is introduced for general shape optimization problems constrained by the incompressible Navier–Stokes equations. The optimization problem is first discretized with a finite element method on an unstructured moving mesh that is implicitly defined without assuming that the computational domain is known and then solved by some one‐shot Lagrange–Newton–Krylov–Schwarz algorithms. In this approach, the shape of the domain, its corresponding finite element mesh, the flow fields and their corresponding Lagrange multipliers are all obtained computationally in a single solve of a nonlinear system of equations. Highly scalable parallel algorithms are absolutely necessary to solve such an expensive system. The one‐level domain decomposition method works reasonably well when the number of processors is not large. Aiming for machines with a large number of processors and robust nonlinear convergence, we introduce a two‐level inexact Newton method with a hybrid two‐level overlapping Schwarz preconditioner. As applications, we consider the shape optimization of a cannula problem and an artery bypass problem in 2D. Numerical experiments show that our algorithm performs well on a supercomputer with over 1000 processors for problems with millions of unknowns. Copyright © 2014 John Wiley & Sons, Ltd.     

A low-order unstructured-mesh approach for computational electromagnetics in the time domain

Maxwell's curl equations in the time domain are solved using an explicit linear finite-element approach implemented on unstructured tetrahedral meshes. For the simulation of scattering problems, a perfectly matched layer is added at the artificial far-field boundary, created by the truncation of the physical domain prior to the numerical solution. The complete solution procedure is parallelized. The computational challenges that are encountered when attempting simulations at higher frequencies suggest that the implementation of a hybrid algorithm could have certain advantages. The hybrid approach adopted uses a combination of the finite-element procedure and the well-known low operation count/low storage finite-difference time-domain method. Examples are included to demonstrate the numerical performance of the techniques that are described.     

Basis mapping methods for forward and inverse problems

This paper describes a novel method for mapping between basis representation of a field variable over a domain in the context of numerical modelling and inverse problems. In the numerical solution of inverse problems, a continuous scalar or vector field over a domain may be represented in different finite‐dimensional basis approximations, such as an unstructured mesh basis for the numerical solution of the forward problem, and a regular grid basis for the representation of the solution of the inverse problem. Mapping between the basis representations is generally lossy, and the objective of the mapping procedure is to minimise the errors incurred. We present in this paper a novel mapping mechanism that is based on a minimisation of the L² or H¹ norm of the difference between the two basis representations. We provide examples of mapping in 2D and 3D problems, between an unstructured mesh basis representative of an FEM approximation, and different types of structured basis including piecewise constant and linear pixel basis, and blob basis as a representation of the inverse basis. A comparison with results from a simple sampling‐based mapping algorithm shows the superior performance of the method proposed here. © 2016 The Authors. International Journal for Numerical Methods in Engineering Published by John Wiley & Sons Ltd.