An embedded mesh method for treating overlapping finite element meshes

A new technique for treating the mechanical interactions of overlapping finite element meshes is presented. Such methods can be useful for numerous applications, for example, fluid–solid interaction with a superposed meshed solid on an Eulerian background fluid grid. In this work, we consider the interaction of two elastic domains: one mesh is the foreground and defines the surface of interaction, the other is a background mesh and is often a structured grid. Many of the previously proposed methods employ surface defined Lagrange multipliers or penalties to enforce the boundary constraints. It has become apparent that these methods will cause mesh locking under certain conditions. Appropriately applied, the Nitsche method can overcome this locking, but, in its canonical form, is generally not applicable to non‐linear materials such as hyperelastics. The relationship between interior point penalty, discontinuous Galerkin and Nitsche's method is well known. Based on this relationship, a nonlinear theory analogous to the Nitsche method is proposed to treat nonlinear materials in an embedded mesh. Here, a discontinuous Galerkin derivative based on a lifting of the interface surface integrals provides a consistent treatment for non‐linear materials and demonstrates good behavior in example problems. Published 2012. This article is a US Government work and is in the public domain in the USA.     

Imposing Dirichlet boundary conditions with Nitsche's method and spline‐based finite elements

A key challenge while employing non‐interpolatory basis functions in finite‐element methods is the robust imposition of Dirichlet boundary conditions. The current work studies the weak enforcement of such conditions for B‐spline basis functions, with application to both second‐ and fourth‐order problems. This is achieved using concepts borrowed from Nitsche's method, which is a stabilized method for imposing constraints on surfaces. Conditions for the stability of the system of equations are derived for each class of problem. Stability parameters in the Nitsche weak form are then evaluated by solving a local generalized eigenvalue problem at the Dirichlet boundary. The approach is designed to work equally well when the grid used to build the splines conforms to the physical boundary of interest as well as to the more general case when it does not. Through several numerical examples, the approach is shown to yield optimal rates of convergence. Copyright © 2010 John Wiley & Sons, Ltd.     

Boundary knot method for some inverse problems associated with the Helmholtz equation

The boundary knot method is an inherently meshless, integration‐free, boundary‐type, radial basis function collocation technique for the solution of partial differential equations. In this paper, the method is applied to the solution of some inverse problems for the Helmholtz equation, including the highly ill‐posed Cauchy problem. Since the resulting matrix equation is badly ill‐conditioned, a regularized solution is obtained by employing truncated singular value decomposition, while the regularization parameter for the regularization method is provided by the L‐curve method. Numerical results are presented for both smooth and piecewise smooth geometry. The stability of the method with respect to the noise in the data is investigated by using simulated noisy data. The results show that the method is highly accurate, computationally efficient and stable, and can be a competitive alternative to existing methods for the numerical solution of the problems. Copyright © 2005 John Wiley & Sons, Ltd.     

Embedded kinematic boundary conditions for thin plate bending by Nitsche's approach

A stabilized variational formulation, based on Nitsche's method for enforcing boundary constraints, leads to an efficient procedure for embedding kinematic boundary conditions in thin plate bending. The absence of kinematic admissibility constraints allows the use of non‐conforming meshes with non‐interpolatory approximations, thereby providing added flexibility in addressing the C¹‐continuity requirements typical of these problems. Work‐conjugate pairs weakly enforce kinematic boundary conditions. The pointwise enforcement of corner deflections is key to good performance in the presence of corners. Stabilization parameters are determined from local generalized eigenvalue problems, guaranteeing coercivity of the discrete bilinear form. The accuracy of the approach is verified by representative computations with bicubic C² B‐splines, exhibiting optimal rates of convergence and robust performance with respect to values of the stabilization parameters. Copyright © 2012 John Wiley & Sons, Ltd.     

A symmetric method for weakly imposing Dirichlet boundary conditions in embedded finite element meshes

In this paper, we propose a way to weakly prescribe Dirichlet boundary conditions in embedded finite element meshes. The key feature of the method is that the algorithmic parameter of the formulation which allows to ensure stability is independent of the numerical approximation, relatively small, and can be fixed a priori. Moreover, the formulation is symmetric for symmetric problems. An additional element-discontinuous stress field is used to enforce the boundary conditions in the Poisson problem. Additional terms are required in order to guarantee stability in the convection–diffusion equation and the Stokes problem. The proposed method is then easily extended to the transient Navier–Stokes equations. Copyright © 2012 John Wiley & Sons, Ltd.     

On methods for stabilizing constraints over enriched interfaces in elasticity

Enriched finite element approaches such as the extended finite element method provide a framework for constructing approximations to solutions of non‐smooth problems. Internal features, such as boundaries, are represented in such methods by using discontinuous enrichment of the standard finite element basis. Within such frameworks, however, imposition of interface constraints and/or constitutive relations can cause unexpected difficulties, depending upon how relevant fields are interpolated on un‐gridded interfaces. This work address the stabilized treatment of constraints in an enriched finite element context. Both the Lagrange multiplier and penalty enforcement of tied constraints for an arbitrary boundary represented in an enriched finite element context can lead to instabilities and artificial oscillations in the traction fields. We demonstrate two alternative variational methods that can be used to enforce the constraints in a stable manner. In a ‘bubble‐stabilized approach,’ fine‐scale degrees of freedom are added over elements supporting the interface. The variational form can be shown to have a similar form to a second approach we consider, Nitsche's method, with the exception that the stabilization terms follow directly from the bubble functions. In this work, we examine alternative variational methods for enforcing a tied constraint on an enriched interface in the context of two‐dimensional elasticity. We examine several benchmark problems in elasticity, and show that only Nitsche's method and the bubble‐stabilization approach produce stable traction fields over internal boundaries. We also demonstrate a novel difference between the penalty method and Nitsche's method in that the latter passes the patch test exactly, regardless of the stabilization parameter's magnitude. Results for more complicated geometries and triple interface junctions are also presented. Copyright © 2008 John Wiley & Sons, Ltd.     

A Trefftz based method for solving Helmholtz problems in semi-infinite domains

The wave based method (WBM), which is based on an indirect Trefftz approach, is a deterministic prediction method posed as an alternative to the element-based methods. It uses wave functions, which are exact solutions of the underlying differential equation, to describe the dynamic field variables. In this way, it can avoid the pollution errors associated with the polynomial element-based approximations. As a consequence, a dense element discretization is no longer required, yielding a smaller numerical system. The resulting enhanced computational efficiency of the WBM as compared to the element-based methods has been proven for the analysis of both bounded and unbounded acoustic problems. This paper extends the applicability of the WBM to semi-infinite domains. An appropriate function set is proposed, together with a calculation procedure for both semi-infinite radiation and scattering problems, and transmission or diffraction problems containing a rigid baffle. The resulting technique is validated on two numerical examples.     

An efficient finite element method for embedded interface problems

A stabilized finite element method based on the Nitsche technique for enforcing constraints leads to an efficient computational procedure for embedded interface problems. We consider cases in which the jump of a field across the interface is given, as well as cases in which the primary field on the interface is given. The finite element mesh need not be aligned with the interface geometry. We present closed‐form analytical expressions for interfacial stabilization terms and simple procedures for accurate flux evaluations. Representative numerical examples demonstrate the effectiveness of the proposed methodology. Copyright © 2008 John Wiley & Sons, Ltd.     

New investigations into the BKM for inverse problems of Helmholtz equation

The boundary knot method (BKM) is an inherent boundary-type meshless collocation method for partial differential equations (PDEs). Using non-singular general solutions, numerical solutions of the PDE can be obtained based on the boundary points. In this paper, we investigate the applications of the BKM to solve Helmholtz problems involving various boundary conditions. We use the effective condition number to investigate the ill-conditioned interpolation system. Different from previous investigations, numerical results in this paper reveal that the BKM is promising in dealing with Helmholtz problems under only partially accessible boundary conditions.     

A bubble‐stabilized finite element method for Dirichlet constraints on embedded interfaces

We examine a bubble‐stabilized finite element method for enforcing Dirichlet constraints on embedded interfaces. By ‘embedded’ we refer to problems of general interest wherein the geometry of the interface is assumed independent of some underlying bulk mesh. As such, the robust imposition of Dirichlet constraints using a Lagrange multiplier field is not trivial. To focus issues, we consider a simple one‐sided problem that is representative of a wide class of evolving‐interface problems. The bulk field is decomposed into coarse and fine scales, giving rise to coarse‐scale and fine‐scale one‐sided sub‐problems. The fine‐scale solution is approximated with bubble functions, permitting static condensation and giving rise to a stabilized form bearing strong analogy with a classical method. Importantly, the method is simple to implement, readily extends to multiple dimensions, obviates the need to specify any free stabilization parameters, and can lead to a symmetric, positive‐definite system of equations. The performance of the method is then examined through several numerical examples. The accuracy of the Lagrange multiplier is compared to results obtained using a local version of the domain integral method. The variational multiscale approach proposed herein is shown to both stabilize the Lagrange multiplier and improve the accuracy of the post‐processed fluxes. Copyright © 2006 John Wiley & Sons, Ltd.     

Comparison of regularization methods for solving the Cauchy problem associated with the Helmholtz equation

In this paper, several boundary element regularization methods, such as iterative, conjugate gradient, Tikhonov regularization and singular value decomposition methods, for solving the Cauchy problem associated to the Helmholtz equation are developed and compared. Regularizing stopping criteria are developed and the convergence, as well as the stability, of the numerical methods proposed are analysed. The Cauchy problem for the Helmholtz equation can be regularized by various methods, such as the general regularization methods presented in this paper, but more accurate results are obtained by classical methods, such as the singular value decomposition and the Tikhonov regularization methods. Copyright © 2004 John Wiley & Sons, Ltd.     

Nitsche's method for coupling non-matching meshes in fluid-structure vibration problems

Nitsche's method [11] is a classical method for imposing essential boundary conditions weakly. Unlike the penalty method, it is consistent with the original differential equation. The strong point of Nitsche's method is that it retains the convergence rate of the underlying finite element method, whereas the standard penalty method either requires a very large penalty parameter, destroying the condition number of the resulting matrix problem, or, in case the condition number is to be retained, is limited to first order energy-norm accuracy. In this paper, we give a formulation of Nitsche's method suitable for the problem of fluid-structure interaction. Numerical examples are given.     

基本解方法解水下刚性目标三维Helmholtz外散射问题

运用基本解方法求解水下刚性目标三维Helmholtz外散射问题。研究了源点位置分布和数目对基本解方法计算结果的影响,比较了最小二乘配点法和等额配点法的计算精度。结果表明,当源点构成的形状与目标边界的形状差异大时,计算精度差,增加源点的数量可提高计算精度,运用较少的源点也可获得令人满意的精度,从而提高计算效率,但源点不宜距离目标边界过远;最小二乘配点法的计算精度较等额配点法高些。     

A Galerkin least-squares finite element method for the two-dimensional Helmholtz equation

In this paper a Galerkin least-squares (GLS) finite element method, in which residuals in least-squares form are added to the standard Galerkin variational equation, is developed to solve the Helmholtz equation in two dimensions. An important feature of GLS methods is the introduction of a local mesh parameter that may be designed to provide accurate solutions with relatively coarse meshes. Previous work has accomplished this for the one-dimensional Helmholtz equation using dispersion analysis. In this paper, the selection of the GLS mesh parameter for two dimensions is considered, and leads to elements that exhibit improved phase accuracy. For any given direction of wave propagation, an optimal GLS mesh parameter is determined using two-dimensional Fourier analysis. In general problems, the direction of wave propagation will not be known a priori. In this case, an optimal GLS parameter is found which reduces phase error for all possible wave vector orientations over elements. The optimal GLS parameters are derived for both consistent and lumped mass approximations. Several numerical examples are given and the results compared with those obtained from the Galerkin method. The extension of GLS to higher-order quadratic interpolations is also presented.     

Parallel multipole implementation of the generalized Helmholtz decomposition for solving viscous flow problems

The evaluation of a domain integral is the dominant bottleneck in the numerical solution of viscous flow problems by vorticity methods, which otherwise demonstrate distinct advantages over primitive variable methods. By applying a Barnes–Hut multipole acceleration technique, the operation count for the integration is reduced from O(N²) to O(NlogN), while the memory requirements are reduced from O(N²) to O(N). The algorithmic parameters that are necessary to achieve such scaling are described. The parallelization of the algorithm is crucial if the method is to be applied to realistic problems. A parallelization procedure which achieves almost perfect scaling is shown. Finally, numerical experiments on a driven cavity benchmark problem are performed. The actual increase in performance and reduction in storage requirements match theoretical predictions well, and the scalability of the procedure is very good. Copyright © 2003 John Wiley Sons, Ltd.     

A domain decomposition method for solving thin film elliptic interface problems with variable coefficients

A domain decomposition method is developed for solving thin film elliptic interface problems with variable coefficients. In this study, the elliptic equation with variable coefficients is discretized using second‐order finite differences while a discrete interface equation is obtained using the immersed interface method in order to obtain a second‐order global accuracy. The obtained linear system is solved using a preconditioned Richardson iteration, which is shown to converge fast when the grid size in the thickness direction is much smaller than the grid sizes in both the length and width directions. To simplify the computation, a domain decomposition algorithm is obtained based on a parallel Gaussian elimination procedure. The method is illustrated by a numerical example. Copyright © 1999 John Wiley & Sons, Ltd.     

An efficient Wave Based Method for solving Helmholtz problems in three-dimensional bounded domains

This paper discusses the use of the Wave Based Method for the analysis of time-harmonic three-dimensional (3D) interior acoustic problems. Conventional element-based prediction methods, such as the Finite Element Method, are most commonly used for these types of problems, but they are restricted to low-frequency applications. The Wave Based Method is an alternative deterministic technique which is based on the indirect Trefftz approach. Up to now, this method's very high computational efficiency has been illustrated mainly for two-dimensional (2D) problem settings, allowing the analysis of problems at higher frequencies. The numerical validation examples presented in this work shows that the enhanced computational efficiency of the Wave Based Method in comparison with conventional element-based methods is kept when the method is extended to 3D case with and without the presence of material damping.     

A two‐level domain decomposition method with accurate interface conditions for the Helmholtz problem

A new and efficient two‐level, non‐overlapping domain decomposition (DD) method is developed for the Helmholtz equation in the two Lagrange multiplier framework. The transmission conditions are designed by utilizing perfectly matched discrete layers (PMDLs), which are a more accurate representation of the exterior Dirichlet‐to‐Neumann map than the polynomial approximations used in the optimized Schwarz method. Another important ingredient affecting the convergence of a DD method, namely, the coarse space augmentation, is also revisited. Specifically, the widely successful approach based on plane waves is modified to that based on interface waves, defined directly on the subdomain boundaries, hence ensuring linear independence and facilitating the estimation of the optimal size for the coarse problem. The effectiveness of both PMDL‐based transmission conditions and interface‐wave‐based coarse space augmentation is illustrated with an array of numerical experiments that include comprehensive scalability studies with respect to frequency, mesh size and the number of subdomains. Copyright © 2015 John Wiley & Sons, Ltd.