A residual‐based finite element method for the Helmholtz equation

A new residual‐based finite element method for the scalar Helmholtz equation is developed. This method is obtained from the Galerkin approximation by appending terms that are proportional to residuals on element interiors and inter‐element boundaries. The inclusion of residuals on inter‐element boundaries distinguishes this method from the well‐known Galerkin least‐squares method and is crucial to the resulting accuracy of this method. In two dimensions and for regular bilinear quadrilateral finite elements, it is shown via a dispersion analysis that this method has minimal phase error. Numerical experiments are conducted to verify this claim as well as test and compare the performance of this method on unstructured meshes with other methods. It is found that even for unstructured meshes this method retains a high level of accuracy. Copyright © 2000 John Wiley & Sons, Ltd.     

Comparison of high‐order curved finite elements

Several finite element techniques used in domains with curved boundaries are discussed and compared, with particular emphasis in two issues: the exact boundary representation of the domain and the consistency of the approximation. The influence of the number of integration points on the accuracy of the computation is also studied. Two‐dimensional numerical examples, solved with continuous and discontinuous Galerkin formulations, are used to test and compare all these methodologies. In every example shown, the recently proposed NURBS‐enhanced finite element method (NEFEM) provides the maximum accuracy for a given spatial discretization, at least one order of magnitude more accurate than classical isoparametric finite element method (FEM). Moreover, NEFEM outperforms Cartesian FEM and p‐FEM, stressing the importance of the geometrical model as well as the relevance of a consistent approximation in finite element simulations. Copyright © 2011 John Wiley & Sons, Ltd.     

Numerical accuracy of a Padé‐type non‐reflecting boundary condition for the finite element solution of acoustic scattering problems at high‐frequency

The present text deals with the numerical solution of two‐dimensional high‐frequency acoustic scattering problems using a new high‐order and asymptotic Padé‐type artificial boundary condition. The Padé‐type condition is easy‐to‐implement in a Galerkin least‐squares (iterative) finite element solver for arbitrarily convex‐shaped boundaries. The method accuracy is investigated for different model problems and for the scattering problem by a submarine‐shaped scatterer. As a result, relatively small computational domains, optimized according to the shape of the scatterer, can be considered while yielding accurate computations for high‐frequencies. Copyright © 2005 John Wiley & Sons, Ltd.     

Phase‐field modelling of solidification microstructure formation using the discontinuous finite element method

This paper presents a discontinuous Galerkin finite element computational methodology for solving the coupled phase‐field and heat diffusion equations to predict the microstructure evolution during solidification. The phase‐field modelling of microstructure formation is briefly discussed. The discontinuous Galerkin finite element formulation for the phase‐field model systems for solidification microstructure formation is described in detail. Numerical stability is performed using the Neumann method. The accuracy of the discontinuous model is verified by the analytic solution of a simple 1‐D solidification problem. Numerical calculations using the discontinuous finite element phase‐field model have been performed for simulating the complex 2‐D and 3‐D dendrite structures formed in supercooled melts and the results are compared well with those in literature using the finite difference methods. Parallel computing algorithm is presented and results show that the minimization of the intercommunication between microprocessors is the key to increase the effectiveness in parallel computing with the discontinuous finite element phase‐field model for solidification microstructure formation. Copyright © 2006 John Wiley & Sons, Ltd.     

A general discontinuous Galerkin method for finite hyperelasticity. Formulation and numerical applications

A discontinuous Galerkin formulation of the boundary value problem of finite‐deformation elasticity is presented. The primary purpose is to establish a discontinuous Galerkin framework for large deformations of solids in the context of statics and simple material behaviour with a view toward further developments involving behaviour or models where the DG concept can show its superiority compared to the continuous formulation. The method is based on a general Hu–Washizu–de Veubeke functional allowing for displacement and stress discontinuities in the domain interior. It is shown that this approach naturally leads to the formulation of average stress fluxes at interelement boundaries in a finite element implementation. The consistency and linearized stability of the method in the non‐linear range as well as its convergence rate are proven. An implementation in three dimensions is developed, showing that the proposed method can be integrated into conventional finite element codes in a straightforward manner. In order to demonstrate the versatility, accuracy and robustness of the method examples of application and convergence studies in three dimensions are provided. Copyright © 2006 John Wiley & Sons, Ltd.     

One‐dimensional dispersion analysis for the element‐free Galerkin method for the Helmholtz equation

The standard finite element method (FEM) is unreliable to compute approximate solutions of the Helmholtz equation for high wave numbers due to the dispersion, unless highly refined meshes are used, leading to unacceptable resolution times. The paper presents an application of the element‐free Galerkin method (EFG) and focuses on the dispersion analysis in one dimension. It shows that, if the basis contains the solution of the homogenized Helmholtz equation, it is possible to eliminate the dispersion in a very natural way while it is not the case for the finite element methods. For the general case, it also shows that it is possible to choose the parameters of the method in order to minimize the dispersion. Finally, theoretical developments are validated by numerical experiments showing that, for the same distribution of nodes, the element‐free Galerkin method solution is much more accurate than the finite element one. Copyright © 2000 John Wiley & Sons, Ltd.     

A discontinuous‐Galerkin‐based immersed boundary method

A numerical method to approximate partial differential equations on meshes that do not conform to the domain boundaries is introduced. The proposed method is conceptually simple and free of user‐defined parameters. Starting with a conforming finite element mesh, the key ingredient is to switch those elements intersected by the Dirichlet boundary to a discontinuous‐Galerkin approximation and impose the Dirichlet boundary conditions strongly. By virtue of relaxing the continuity constraint at those elements, boundary locking is avoided and optimal‐order convergence is achieved. This is shown through numerical experiments in reaction–diffusion problems. Copyright © 2008 John Wiley & Sons, Ltd.     

High‐order finite elements applied to the discrete Boltzmann equation

A discontinuous Galerkin approach for solving the discrete Boltzmann equation is presented, allowing to compute approximate solutions for fluid flow problems. Based on a two‐dimensional high‐order finite element and an explicit Euler time stepping scheme, the D2Q9 model is discretized and the results are compared to the exact solution of the Navier–Stokes equation. Four numerical examples are considered, including stationary and instationary problems with curved boundaries. It is demonstrated that the proposed method allows to obtain the desired, highly efficient exponential convergence. Copyright © 2006 John Wiley & Sons, Ltd.     

An element‐free method for in‐plane notch problems with anisotropic materials

This study developed an element‐free Galerkin method (EFGM) to simulate notched anisotropic plates containing stress singularities at the notch tip. Two‐dimensional theoretical complex displacement functions are first deduced into the moving least‐squares interpolation. The interpolation functions and their derivatives are then determined to calculate the nodal stiffness using the Galerkin method. In the numerical validation, an interface layer of the EFGM is used to combine the mesh between the traditional finite elements and the proposed singular notch EFGM. The H‐integral determined from finite element analyses with a very fine mesh is used to validate the numerical results of the proposed method. The comparisons indicate that the proposed method obtains more accurate results for the displacement, stress, and energy fields than those determined from the standard finite element method. Copyright © 2013 John Wiley & Sons, Ltd.     

The simple boundary element method for multiple cracks in functionally graded media governed by potential theory: a three‐dimensional Galerkin approach

The simple boundary element method consists of recycling existing codes for homogeneous media to solve problems in non‐homogeneous media while maintaining a purely boundary‐only formulation. Within this scope, this paper presents a ‘simple’ Galerkin boundary element method for multiple cracks in problems governed by potential theory in functionally graded media. Steady‐state heat conduction is investigated for thermal conductivity varying either parabolically, exponentially, or trigonometrically in one or more co‐ordinates. A three‐dimensional implementation which merges the dual boundary integral equation technique with the Galerkin approach is presented. Special emphasis is given to the treatment of crack surfaces and boundary conditions. The test examples simulated with the present method are verified with finite element results using graded finite elements. The numerical examples demonstrate the accuracy and efficiency of the present method especially when multiple interacting cracks are involved. Copyright © 2005 John Wiley & Sons, Ltd.     

A new stabilized finite element method for reaction–diffusion problems: The source‐stabilized Petrov–Galerkin method

This paper proposes a new stabilized finite element method to solve singular diffusion problems described by the modified Helmholtz operator. The Galerkin method is known to produce spurious oscillations for low diffusion and various alternatives were proposed to improve the accuracy of the solution. The mostly used methods are the well‐known Galerkin least squares and Galerkin gradient least squares (GGLS). The GGLS method yields the exact nodal solution in the one‐dimensional case and for a uniform mesh. However, the behavior of the method deteriorates slightly in the multi‐dimensional case and for non‐uniform meshes. In this work we propose a new stabilized finite element method that leads to improved accuracy for multi‐dimensional problems. For the one‐dimensional case, the new method leads to the same results as the GGLS method and hence provides exact nodal solutions to the problem on uniform meshes. The proposed method is a Galerkin discretization used to solve a modified equation that includes a term depending on the gradient of the original partial differential equation. Copyright © 2008 John Wiley & Sons, Ltd.     

A quadtree data structure for the coupled finite‐element/element‐free Galerkin method

In the present paper, we derive an efficient data structure for the organization of the nodes in the coupled finite element/element‐free Galerkin method. With respect to its implementation, we compare various approaches of recursive spatial discretizations that facilitate most flexible handling of the nodes. The goal of the paper is to refine the implementation issues of the data structure which is fundamental to the element‐free Galerkin method and thus to speed‐up this otherwise computationally rather expensive meshfree method. Copyright © 2001 John Wiley & Sons, Ltd.     

Blending in the extended finite element method by discontinuous Galerkin and assumed strain methods

In the extended finite element method (XFEM), errors are caused by parasitic terms in the approximation space of the blending elements at the edge of the enriched subdomain. A discontinuous Galerkin (DG) formulation is developed, which circumvents this source of error. A patch‐based version of the DG formulation is developed, which decomposes the domain into enriched and unenriched subdomains. Continuity between patches is enforced with an internal penalty method. An element‐based form is also developed, where each element is considered a patch. The patch‐based DG is shown to have similar accuracy to the element‐based DG for a given discretization but requires significantly fewer degrees of freedom. The method is applied to material interfaces, cracks and dislocation problems. For the dislocations, a contour integral form of the internal forces that only requires integration over the patch boundaries is developed. A previously developed assumed strain (AS) method is also developed further and compared with the DG method for weak discontinuities and linear elastic cracks. The DG method is shown to be significantly more accurate than the standard XFEM for a given element size and to converge optimally, even where the standard XFEM does not. The accuracy of the DG method is similar to that of the AS method but requires less application‐specific coding. Copyright © 2007 John Wiley & Sons, Ltd.     

Eulerian formulation using stabilized finite element method for large deformation solid dynamics

This paper describes an Eulerian formulation for large deformation solid dynamics. In the present Eulerian approach, an advective equation is solved using the Stream‐Upwind/Petrov–Galerkin finite element method. The Eulerian finite element method is applied to path‐dependent solid analyses such as impact bar and ductile necking problems. These computational results using the Eulerian finite element method are compared with the results obtained from using the Lagrangian finite element method and an Eulerian formulation based on a finite difference method. Copyright © 2007 John Wiley & Sons, Ltd.     

A three‐dimensional finite element method with arbitrary polyhedral elements

The ‘variable‐element‐topology finite element method’ (VETFEM) is a finite‐element‐like Galerkin approximation method in which the elements may take arbitrary polyhedral form. A complete development of the VETFEM is given here for both two and three dimensions. A kinematic enhancement of the displacement‐based formulation is also given, which effectively treats the case of near‐incompressibility. Convergence of the method is discussed and then illustrated by way of a 2D problem in elastostatics. Also, the VETFEM's performance is compared to that of the conventional FEM with eight‐node hex elements in a 3D finite‐deformation elastic–plastic problem. The main attraction of the new method is its freedom from the strict rules of construction of conventional finite element meshes, making automatic mesh generation on complex domains a significantly simpler matter. Copyright © 2006 John Wiley & Sons, Ltd.     

On a two‐level finite element method for the incompressible Navier–Stokes equations

We consider the Galerkin finite element method for the incompressible Navier–Stokes equations in two dimensions, where the finite‐dimensional space(s) employed consist of piecewise polynomials enriched with residual‐free bubble functions. To find the bubble part of the solution, a two‐level finite element method (TLFEM) is described and its application to the Navier–Stokes equation is displayed. Numerical solutions employing the TLFEM are presented for three benchmark problems. We compare the numerical solutions using the TLFEM with the numerical solutions using a stabilized method. Copyright © 2001 John Wiley & Sons, Ltd.     

Buckling of folded plate structures subjected to partial in‐plane edge loads by the FSDT meshfree Galerkin method

This paper presents a meshfree Galerkin method that is based on the first‐order shear deformation theory (FSDT) to study the elastic buckling behaviour of stiffened and un‐stiffened folded plates under partial in‐plane edge loads. The un‐stiffened folded plates are modelled as assemblies of flat plates. The stiffness and initial stress matrices of the flat plates are derived by the meshfree Galerkin method. A treatment is implemented to modify the stiffness and initial stress matrices, and the matrices are then superposed to obtain the stiffness and initial stress matrix of the entire folded plate. The analytical process for stiffened folded plates is similar, except that the effects of the stiffeners must be taken into account. Because no mesh is required, the proposed method is superior for studying problems that would involve remeshing in the finite element method. Several examples are employed to show the convergence and accuracy of the proposed method. The results obtained show good agreement with the results computed from the finite element analysis software ANSYS. Copyright © 2005 John Wiley & Sons, Ltd.     

Locking in the incompressible limit for the element‐free Galerkin method

Volumetric locking (locking in the incompressible limit) for linear elastic isotropic materials is studied in the context of the element‐free Galerkin method. The modal analysis developed here shows that the number of non‐physical locking modes is independent of the dilation parameter (support of the interpolation functions). Thus increasing the dilation parameter does not suppress locking. Nevertheless, an increase in the dilation parameter does reduce the energy associated with the non‐physical locking modes; thus, in part, it alleviates the locking phenomena. This is shown for linear and quadratic orders of consistency. Moreover, the biquadratic order of consistency, as in finite elements, improves the locking behaviour. Although more locking modes are present in the element‐free Galerkin method with quadratic consistency than with standard biquadratic finite elements. Finally, numerical examples are shown to validate the modal analysis. In particular, the conclusions of the modal analysis are also confirmed in an elastoplastic example. Copyright © 2001 John Wiley & Sons, Ltd.     

A stabilized volume‐averaging finite element method for flow in porous media and binary alloy solidification processes

A stabilized equal‐order velocity–pressure finite element algorithm is presented for the analysis of flow in porous media and in the solidification of binary alloys. The adopted governing macroscopic conservation equations of momentum, energy and species transport are derived from their microscopic counterparts using the volume‐averaging method. The analysis is performed in a single domain with a fixed numerical grid. The fluid flow scheme developed includes SUPG (streamline‐upwind/Petrov–Galerkin), PSPG (pressure stabilizing/Petrov–Galerkin) and DSPG (Darcy stabilizing/Petrov–Galerkin) stabilization terms in a variable porosity medium. For the energy and species equations a classical SUPG‐based finite element method is employed. The developed algorithms were tested extensively with bilinear elements and were shown to perform stably and with nearly quadratic convergence in high Rayleigh number flows in varying porosity media. Examples are shown in natural and double diffusive convection in porous media and in the directional solidification of a binary‐alloy. Copyright © 2004 John Wiley & Sons, Ltd.     

Continuously deforming finite elements for the solution of parabolic problems,with and without phase change

A number of transport problems are complicated by the presence of physically important transition zones where quantities exhibit steep gradients and special numerical care is required. When the location of such a transition zone changes as the solution evolves through time, use of a deforming numerical mesh is appropriate in order to preserve the proper numerical features both within the transition zone and at its boundaries. A general finite element solution method is described wherein the elements are allowed to deform continuously, and the effects of this deformation are accounted for exactly. The method is based on the Galerkin approximation in space, and uses finite difference approximations for the time derivatives. In the absence of element deformation, the method reduces to the conventional Galerkin formulation. The method is applied to the two-phase Stefan problem associated with the melting and solidification of A substance. The interface between the solid and liquid phase form an internal moving boundary, and latent heat effects are accounted for in the associated boundary condition. By allowing continuous mesh deformation, as dictated by this boundary condition, the moving boundary always lies on element boundaries. This circumvents the difficulties inherent in interpolation of parameters and dependent variables across regions where those quantities change abruptly. Basis functions based on Hermite polynomials are used, to allow exact specification of the flux-latent heat balance condition at the phase boundary. Analytic solutions for special cases provide tests of the method.