A comparison of the meshless RBF collocation method with finite element and boundary element methods in neutron diffusion calculations

Efficient coupling of finite elements and boundary elements---adaptive procedures and preconditioners

The fluid-structure interaction technique provides a paradigm for solving scattering from elastic targets embedded in a fluid by a combination of finite and boundary element methods. In this technique, the finite element method is used to compute the target’s elastic response and the boundary element method with the appropriate Green’s function is used to compute the field in the exterior medium. The two methods are coupled at the surface of the target by imposing the continuity of pressure and normal displacement. This results in a self-consistent boundary element method that can be used to compute the scattered field anywhere in the surrounding environment. The method reduces a finite element problem to a boundary element one with drastic reduction in the number of unknowns, which translates to a significant reduction in numerical cost. In this talk, the method is extended to compute scattering from multiple targets by self-consistently accounting for all interactions between them. The model allows to identify block matrices responsible for the interaction between targets, which proves useful in many applications. The model is tested by comparing its results with those measured involving two aluminum cylinders one of which is excited by modulated radiation pressure.

Numerical methods can be characterized by the basis used, the criterion of fit used, and the approximations to the variational formulation which must be made to implement the numerical approximation. Finite element methods usually use a polynomial basis and the criterion of fit is a domain integral variational statement. The boundary element methods usually use a free-space Green’s function as a basis. The criterion of it is usually a collocation method for satisfying a boundary integral equation for discrete specified boundary conditions. For both the finite element and boundary element methods, discretization, numerical mapping, and integration procedures are potential sources of significant error. The characteristics of structural acoustic applications of both the finite element and boundary element methods will be identified and contrasted to analytical methods. It will be demonstrated that numerical methods and analytical methods have different strengths and weaknesses and that the methods should be considered complementary.<ep;balance;l>

In this paper a hybrid, finite element--boundary element method which can be used to solve for particle advection-diffusion in infinite domains with variable advective fields is presented. In previous work either boundary element, finite element, or difference methods have been used to solve for particle motion in advective-diffusive domains. These methods have a number of limitations. Due to the complexity of computing spatially dependent Green`s functions, the boundary element method is limited to domains containing only constant advective fields, and due to their inherent formulation, finite element and finite difference methods are limited to only domains of finite spatial extent. Thus, finite element and finite difference methods are limited to finite space problems for which the boundary element method is not, and the boundary element method is limited to constant advection field problems for which finite element and finite difference methods are not. In this paper it is proposed to split a domain into two sub-domains, and for each of these sub domains, apply the appropriate solution method; thereby, producing a method for the total infinite space, variable advective field domain.

In this paper a hybrid, finite element--boundary element method which can be used to solve for particle advection-diffusion in infinite domains with variable advective fields is presented. In previous work either boundary element, finite element, or difference methods have been used to solve for particle motion in advective-diffusive domains. These methods have a number of limitations. Due to the complexity of computing spatially dependent Green`s functions, the boundary element method is limited to domains containing only constant advective fields, and due to their inherent formulation, finite element and finite difference methods are limited to only domains of finite spatial extent. Thus, finite element and finite difference methods are limited to finite space problems for which the boundary element method is not, and the boundary element method is limited to constant advection field problems for which finite element and finite difference methods are not. In this paper it is proposed to split a domain into two sub-domains, and for each of these sub domains, apply the appropriate solution method; thereby, producing a method for the total infinite space, variable advective field domain.

Coupled finite element and boundary element method in electromagnetics

A 3D hybrid BE–FE solution to the forward problem of electrical impedance tomography

5R3. Scaled Boundary Finite Element Method. - JP Wolf (Swiss Fed Inst of Tech, Lausanne, Switzerland). Wiley, W Sussex, UK. 2003. 361 pp. ISBN 0-471-48682-5. $130.00.Reviewed by Long-Yuan Li (Dept of Civil Eng, Aston Univ, Aston Triangle, Birmingham, B4 7ET, UK).This book describes a fundamental solution-less boundary element method, based on finite elements. The method combines the advantages of both the finite and boundary element methods as the finite element discretization in the method is restricted to the circumferential direction while in the radial direction it uses a scaling procedure to obtain an analytical solution. The method can be used to analyze any bounded and unbounded media governed by linear elliptic, parabolic, and hyperbolic partial differential equations. The book is based on the research and development performed recently by the author and his colleagues. It is a unique research book that presents the development of new numerical procedures that can overcome some difficulties that appear when using the finite or boundary element method. The book contains 26 chapters, 4 appendices, high quality figures, and a good subject index. References are provided in the alpha beta order, which are listed at the end of the chapters. The first two chapters provide a brief introduction of numerical procedures and features of the finite element method, boundary element method, and scaled boundary finite element method. More details of the concepts of the scaled boundary finite element method and its applications in model problems and two- and three-dimensional elastodynamic, static’s and diffusion problems are presented in Part I and II. Part I addresses the model problem, which contains 12 chapters (Chapters 3-14). Chapter 3 addresses the concepts of scaled boundary transformation of geometry and similarity. Chapter 4 gives the definition of a model problem. Two derivations of scaled boundary finite element equations are presented. In Chapter 5 the weighted-residual technique is used, and the other in Chapter 6 uses the similarity and finite element assemblage. Chapter 7 discusses the analytical solution of the scalar scaled boundary finite element equations. Chapters 8-12 discuss the solution procedures of the scaled boundary finite element equations in displacement and in dynamic stiffness for bounded and unbounded media. In Chapter 13, implementation issues are discussed, which also apply to the general matrix equations. Chapter 14 gives the conclusions related to the model problem. At the end of Part I, four short appendices are provided, leading to deeper insight into certain aspects of the model problem, and providing a link to the generalization of two- and three-dimensional static’s, elastodynamics and diffusion in Part II. Appendix A deals with solid modeling, Appendix B discusses the analysis in the frequency domain, Appendix C establishes the equations of motion of a dynamic unbounded medium-structure interaction problem using the properties calculated in the Model Problem, and Appendix D describes the early historical development leading up to the scaled boundary finite element method. Part II has 12 chapters (Chapters 15 to 26), which develops all aspects of the current state of the art of the scaled boundary finite element method. Following the derivation of the fundamental equations based on the scaled boundary transformation (described in Chapter 15), the solution procedures for static’s and dynamics in the frequency and time domains, both numerically and analytically, for bounded and unbounded media are developed in Chapters 16-22, respectively. Two- and three-dimensional examples in elastodynamics and diffusion for bounded and unbounded media are discussed in Chapters 23 and 24. Based on the stress recovery technique error estimation and adaptivity are discussed in Chapter 25. Chapter 26 contains concluding remarks and addresses restrictive properties of the novel method and suggestions for future research. In summary, Scaled Boundary Finite Element Method, is a self-contained, well-presented advanced textbook. It is suitable for research students and for the personal bookshelves of research investigators working in the field of computational engineering sciences. It can also be a useful reference in libraries.

Three dimensional image reconstruction for multi-modality optical spectroscopy systems needs computationally efficient forward solvers with minimum meshing complexity, while allowing the flexibility to apply spatial constraints. Existing models based on the finite element method (FEM) require full 3D volume meshing to incorporate constraints related to anatomical structure via techniques such as regularization. Alternate approaches such as the boundary element method (BEM) require only surface discretization but assume homogeneous or piece-wise constant domains that can be limiting. Here, a coupled finite element-boundary element method (coupled FE-BEM) approach is demonstrated for modeling light diffusion in 3D, which uses surfaces to model exterior tissues with BEM and a small number of volume nodes to model interior tissues with FEM. Such a coupled FE-BEM technique combines strengths of FEM and BEM by assuming homogeneous outer tissue regions and heterogeneous inner tissue regions. Results with FE-BEM show agreement with existing numerical models, having RMS differences of less than 0.5 for the logarithm of intensity and 2.5 degrees for phase of frequency domain boundary data. The coupled FE-BEM approach can model heterogeneity using a fraction of the volume nodes (4-22%) required by conventional FEM techniques. Comparisons of computational times showed that the coupled FE-BEM was faster than stand-alone FEM when the ratio of the number of surface to volume nodes in the mesh (Ns/Nv) was less than 20% and was comparable to stand-alone BEM ( ± 10%).

The postbuckling behavior of thin plates under combined loads is studied in this paper by using a mixed boundary element and finite element method. The transverse and the in-plane deformation of the plates are analyzed by the boundary element method and the finite element method, respectively. Spline functions were used as the interpolation functions and shape functions in the solution of both methods. A quadratic rectangular spline element is adopted in the finite element procedure. Numerical results are given for typical problems to show the effectiveness of the proposed approach. The possibilities to extend the method developed in this paper to more complicated postbuckling problems are discussed in the concluding section.

The prediction of stresses and displacements around tunnels buried deep within the earth is an important class of geomechanics problems. The material behavior immediately surrounding the tunnel is typically nonlinear. The surrounding mass, even if it is nonlinear, can usually be characterized by a simple linear elastic model. The finite element method is best suited for modeling nonlinear materials of limited volume, while the boundary element method is well suited for modeling large volumes of linear elastic material. A computational scheme that couples the finite element and boundary element methods would seem particularly useful for geomechanics problems. A variety of coupling schemes have been proposed, but they rely on direct solution methods. Direct solution techniques have large storage requirements that become cumbersome for large-scale three-dimensional problems. An alternative to direct solution methods is iterative solution techniques. A scheme has been developed for coupling the finite element and boundary element methods that uses an iterative solution method. This report shows that this coupling scheme is valid for problems where nonlinear material behavior occurs in the finite element region.

In this work, formulation of a 2D fully‐coupled finite element method (FEM)/boundary element method (BEM) to simulate the measurements of sound transmission loss of sandwich panels is presented. Specifically, the structural behavior of the sandwich panels, based on a consistent higher‐order theory, is implemented using finite element method (FEM), and the reverberant/anechoic chambers are accessed by boundary element method (BEM). The coupling between the structure and the acoustic medium is achieved by assuming the continuity of the normal velocities at the interface. The absorption of the receiving anechoic chamber is calibrated by comparing the numerically‐predicted sound pressure level difference between the two chambers with the Sewell’s expression for the forced transmission. The obtained correction factors are then used without any modification to predict transmission loss of other sandwich panels with different dimensions and material properties. Numerical examples are presented to validate the numerical procedure. Compared with traditional finite element approach, the proposed hybrid method provides more computation efficiency, and, hence, can be used to study acoustic behavior of sandwich panels at higher frequencies.

Stress and deflection analysis of spur gears are presented by coupling finite and boundary element methods. Single-tooth, three-teeth, and whole-body spur gear models are solved using finite element methods in ANSYS (a finite element package). In addition to these models, two different whole-body models are developed using a coupling program written in Fortran 95. Keyway effects are included in the whole-body and coupling models. In each model, the stresses along the critical sections and deflections of midline of loaded tooth are investigated under the action of driving force. The variations of stresses and deflections are also represented at the critical points when the driving force application point is changing between tip and pitch. The effect of keyway position on stress and deflections is also presented. The results are compared with each other and ANSYS.

The Poisson--Boltzmann equation is widely used to model electrostatics in molecular systems. Available software packages solve it using finite difference, finite element, and boundary element methods, where the latter is attractive due to the accurate representation of the molecular surface and partial charges, and exact enforcement of the boundary conditions at infinity. However, the boundary element method is limited to linear equations and piecewise constant variations of the material properties. In this work, we present a scheme that couples finite and boundary elements for the Poisson--Boltzmann equation, where the finite element method is applied in a confined {\it solute} region, and the boundary element method in the external {\it solvent} region. As a proof-of-concept exercise, we use the simplest methods available: Johnson--N\'ed\'elec coupling with mass matrix and diagonal preconditioning, implemented using the Bempp-cl and FEniCSx libraries via their Python interfaces. We showcase our implementation by computing the polar component of the solvation free energy of a set of molecules using a constant and a Gaussian-varying permittivity. We validate our implementation against the finite difference code APBS (to 0.5\%), and show scaling from protein G B1 (955 atoms) up to immunoglobulin G (20\,148 atoms). For small problems, the coupled method was efficient, outperforming a purely boundary integral approach. For Gaussian-varying permittivities, which are beyond the applicability of boundary elements alone, we were able to run medium to large sized problems on a single workstation. Development of better preconditioning techniques and the use of distributed memory parallelism for larger systems remains an area for future work. We hope this work will serve as inspiration for future developments for molecular electrostatics with implicit solvent models.

The Poisson-Boltzmann equation is widely used to model electrostatics in molecular systems. Available software packages solve it using finite difference, finite element, and boundary element methods, where the latter is attractive due to the accurate representation of the molecular surface and partial charges, and exact enforcement of the boundary conditions at infinity. However, the boundary element method is limited to linear equations and piecewise constant variations of the material properties. In this work, we present a scheme that couples finite and boundary elements for the linearised Poisson-Boltzmann equation, where the finite element method is applied in a confined solute region and the boundary element method in the external solvent region. As a proof-of-concept exercise, we use the simplest methods available: Johnson-Nédélec coupling with mass matrix and diagonal preconditioning, implemented using the Bempp-cl and FEniCSx libraries via their Python interfaces. We showcase our implementation by computing the polar component of the solvation free energy of a set of molecules using a constant and a Gaussian-varying permittivity. As validation, we compare against well-established finite difference solvers for an extensive binding energy data set, and with the finite difference code APBS (to 0.5%) for Gaussian permittivities. We also show scaling results from protein G B1 (955 atoms) up to immunoglobulin G (20,148 atoms). For small problems, the coupled method was efficient, outperforming a purely boundary integral approach. For Gaussian-varying permittivities, which are beyond the applicability of boundary elements alone, we were able to run medium to large-sized problems on a single workstation. The development of better preconditioning techniques and the use of distributed memory parallelism for larger systems remains an area for future work. We hope this work will serve as inspiration for future developments that consider space-varying field parameters, and mixed linear-nonlinear schemes for molecular electrostatics with implicit solvent models.