Interface dynamic stiffness matrix approach for three‐dimensional transient multi‐region boundary element analysis

A novel substructuring method is developed for the coupling of boundary element and finite element subdomains in order to model three‐dimensional multi‐region elastodynamic problems in the time domain. The proposed procedure is based on the interface stiffness matrix approach for static multi‐region problems using variational principles together with the concept of Duhamel integrals. Unit impulses are applied at the boundary of each region in order to evaluate the impulse response matrices of the Duhamel (convolution) integrals. Although the method is not restricted to a special discretization technique, the regions are discretized using the boundary element method combined with the convolution quadrature method. This results in a time‐domain methodology with the advantages of performing computations in the Laplace domain, which produces very accurate and stable results as verified on test examples. In addition, the assembly of the boundary element regions and the coupling to finite elements are greatly simplified and more efficient. Finally, practical applications in the area of soil–structure interaction and tunneling problems are shown. Copyright © 2009 John Wiley & Sons, Ltd.     

A new boundary element method for resonance of an acoustic enclosure

Abstract

This paper presents a new boundary element formulation in which the eigenvalue appears outside the integral operator, which distinguishes it from the Helmholtz integral equation. Thus, the formation of global matrices need only be assembled once. Since the kernel of the operator used in the new formulation is real‐valued, all calculations can be carried out in a much simpler way in the real domain. The complex acoustic pressure amplitude is considered herein to deivate by a certain amount from a harmonic function. It is an important contribution that an exact relation between the deviator and the complex acoustic pressure amplitude is constructed locally and thus no more approximations are introduced except conventional boundary discretizations. Several examples are given to illustrate the feasibility of an accurate, effective prediction of resonance.     

A fully symmetric multi‐zone Galerkin boundary element method

This paper examines the efficient integration of a Symmetric Galerkin Boundary Element Analysis (SGBEA) method with multi‐zone resulting in a fully symmetric Galerkin multi‐zone formulation. In a previous approach, a Galerkin multi‐zone method was developed where the interfacial nodes are assigned degrees of freedom globally so that the displacement and traction continuity across the zonal interfaces are addressed directly. However, the method was only block symmetric. In the present paper, two new approaches are derived. In the first approach, the degrees of freedom for a particular zone are assigned locally, independent of the other zones. The usual linear set of equations, from the symmetric Galerkin approach, are augmented with an additional set of equations generated by the Galerkin form of hypersingular boundary integrals along the interfaces. Zonal continuity is imposed externally through Lagrange's constraints. This approach is also only block symmetric. The second approach derived from the first, uses the continuity constraints at the zonal assembly level to achieve full symmetry. These methods are compared to collocation multi‐zone and an earlier formulation, on two elasticity problems from the literature. It was found that the second method is much faster than the collocation method for medium to large scale problems, primarily due to its complete symmetry. It is also observed that these methods spend marginally more time on integration than the previous Galerkin multi‐zone method but are better suited to parallel processing. Copyright © 1999 John Wiley & Sons, Ltd.     

An efficient complex‐frequency shifted‐perfectly matched layer for second‐order acoustic wave equation

In the context of simulations of wave propagations in unbounded domain, absorbing boundary conditions are often used to truncate the simulation domain to a finite space. Perfectly matched layer (PML) has proven to be an excellent absorbing boundary conditions. However, as this technique was primarily designed for the first‐order equation system, it cannot be applied to the second‐order equation system directly. In this paper, based on a complex‐coordinate stretching technique, we developed a novel, efficient auxiliary‐differential equation form of the complex‐frequency shifted‐PML for the second‐order equation system. This facilitates the use of complex‐frequency shifted‐PML in acoustic simulations based upon wave equations of second‐order form. Compared with previous state‐of‐the‐art methods, the proposed one has the advantage of simpler implementation. It is an unsplit‐field scheme that can be extended to higher‐order discretization schemes conveniently. Numerical results from both homogeneous and heterogeneous computational domains are provided to illustrate the validity of the method. Copyright © 2013 John Wiley & Sons, Ltd.     

Simulating deformation of objects with multi‐materials using boundary element method

In this paper, an algorithm for simulating the elastostatic deformation of multiple objects with different material properties using boundary element method is introduced. By tessellating the surface of a geometric model into elements and classifying all the element nodes into different groups with different attributes, and partitioning the stiffness matrix into several sub‐matrices according to these attributes, a compact expression for the unknown variables is obtained. Comparing with the direct matrix inversion method, the dimension of the system matrix in the expression has been effectively reduced. Besides, this expression shows that the deformation of a multi‐component object can be simulated in a way similar to that of a single‐component object. The capacitance method is adopted for evaluating the deformation of the object. Experimental results illustrate that the proposed method is practical and efficient. Copyright © 2007 John Wiley & Sons, Ltd.     

A new variable‐order singular boundary element for two‐dimensional stress analysis

A new variable‐order singular boundary element for two‐dimensional stress analysis is developed. This element is an extension of the basic three‐node quadratic boundary element with the shape functions enriched with variable‐order singular displacement and traction fields which are obtained from an asymptotic singularity analysis. Both the variable order of the singularity and the polar profile of the singular fields are incorporated into the singular element to enhance its accuracy. The enriched shape functions are also formulated such that the stress intensity factors appear as nodal unknowns at the singular node thereby enabling direct calculation instead of through indirect extrapolation or contour‐integral methods. Numerical examples involving crack, notch and corner problems in homogeneous materials and bimaterial systems show the singular element's great versatility and accuracy in solving a wide range of problems with various orders of singularities. The stress intensity factors which are obtained agree very well with those reported in the literature. Copyright © 2002 John Wiley & Sons, Ltd.     

An h‐hierarchical adaptive procedure for the scaled boundary finite‐element method

The scaled boundary finite‐element method (a novel semi‐analytical method for solving linear partial differential equations) involves the solution of a quadratic eigenproblem, the computational expense of which rises rapidly as the number of degrees of freedom increases. Consequently, it is desirable to use the minimum number of degrees of freedom necessary to achieve the accuracy desired. Stress recovery and error estimation techniques for the method have recently been developed. This paper describes an h‐hierarchical adaptive procedure for the scaled boundary finite‐element method. To allow full advantage to be taken of the ability of the scaled boundary finite‐element method to model stress singularities at the scaling centre, and to avoid discretization of certain adjacent segments of the boundary, a sub‐structuring technique is used. The effectiveness of the procedure is demonstrated through a set of examples. The procedure is compared with a similar h‐hierarchical finite element procedure. Since the error estimators in both cases evaluate the energy norm of the stress error, the computational cost of solutions of similar overall accuracy can be compared directly. The examples include the first reported direct comparison of the computational efficiency of the scaled boundary finite‐element method and the finite element method. The scaled boundary finite‐element method is found to reduce the computational effort considerably. Copyright © 2002 John Wiley & Sons, Ltd.     

Isogeometric FEM‐BEM coupled structural‐acoustic analysis of shells using subdivision surfaces

We introduce a coupled finite and boundary element formulation for acoustic scattering analysis over thin‐shell structures. A triangular Loop subdivision surface discretisation is used for both geometry and analysis fields. The Kirchhoff‐Love shell equation is discretised with the finite element method and the Helmholtz equation for the acoustic field with the boundary element method. The use of the boundary element formulation allows the elegant handling of infinite domains and precludes the need for volumetric meshing. In the present work, the subdivision control meshes for the shell displacements and the acoustic pressures have the same resolution. The corresponding smooth subdivision basis functions have the C¹ continuity property required for the Kirchhoff‐Love formulation and are highly efficient for the acoustic field computations. We verify the proposed isogeometric formulation through a closed‐form solution of acoustic scattering over a thin‐shell sphere. Furthermore, we demonstrate the ability of the proposed approach to handle complex geometries with arbitrary topology that provides an integrated isogeometric design and analysis workflow for coupled structural‐acoustic analysis of shells.     

Fast frequency sweep computations using a multi‐point Padé‐based reconstruction method and an efficient iterative solver

Problems of the form Z (σ) u (σ)= f (σ), where Z is a given matrix, f is a given vector, and σ is a circular frequency or circular frequency‐related parameter arise in many applications including computational structural and fluid dynamics, and computational acoustics and electromagnetics. The straightforward solution of such problems for fine increments of σ is computationally prohibitive, particularly when Z is a large‐scale matrix. This paper discusses an alternative solution approach based on the efficient computation of u and its successive derivatives with respect to σ at a few sample values of this parameter, and the reconstruction of the solution u (σ) in the frequency band of interest using multi‐point Padé approximants. This computational methodology is illustrated with applications from structural dynamics and underwater acoustic scattering. In each case, it is shown to reduce the CPU time required by the straightforward approach to frequency sweep computations by two orders of magnitude. Copyright © 2006 John Wiley & Sons, Ltd.     

An indirect boundary element method for three-dimensional dynamic fracture mechanics

Indirect boundary element methods (fictitious load and displacement discontinuity) have been developed for the analysis of three-dimensional elastostatic and elastodynamic fracture mechanics problems. A set of boundary integral equations for fictitious loads and displacement discontinuities have been derived. The stress intensity factors were obtained by the stress equivalent method for static loading. For dynamic loading the problem was studied in Laplace transform space where the numerical calculation procedure, for the stress intensity factor K_I(p), is the same: as that for the static problem. The Durbin inversion method for Laplace transforms was used to obtain the stress intensity factors in the time domain K_I(t). Results of this analysis are presented for a square bar, with either a rectangular or a circular crack, under static and dynamic loads.     

A novel hybrid FS‐FEM/SEA for the analysis of vibro‐acoustic problems

Predicting the frequency response of a complex vibro‐acoustic system becomes extremely difficult in the mid‐frequency regime. In this work, a novel hybrid face‐based smoothed finite element method/statistical energy analysis (FS‐FEM/SEA) method is proposed, aiming to further improve the accuracy of ‘mid‐frequency’ predictions. According to this approach, the whole vibro‐acoustic system is divided into a combination of a plate subsystem with statistical behaviour and an acoustic cavity subsystem with deterministic behaviour. The plate subsystem is treated using the recently developed FS‐FEM, and the cavity subsystem is dealt with using the SEA. These two different types of subsystems can be coupled and interacted through the so‐called diffuse field reciprocity relation. The ensemble average response of the system is calculated, and the uncertainty is confined and treated in the SEA subsystems. The use of FS‐FEM ‘softens’ the well‐known ‘overly stiff’ behaviour in the standard FEM and reduces the inherent numerical dispersion error. The proposed FS‐FEM/SEA approach is verified and its features are examined by various numerical examples. Copyright © 2015 John Wiley & Sons, Ltd.     

An efficient dual boundary element technique for a two-dimensional fracture problem with multiple cracks

An efficient dual boundary element technique for the analysis of a two-dimensional finite body with multiple cracks is established. In addition to the displacement integral equation derived for the outer boundary, since the relative displacement of the crack surfaces is adopted in the formulation, only the traction integral equation is established on one of the crack surfaces. For each crack, a virtual boundary is devised and connected to one of the crack surfaces to construct a closed integral path. The rigid body translation for the domain enclosed by the closed integral path is then employed for evaluating the hypersingular integral. To solve the dual displacement/traction integral equations simultaneously, the constant and quadratic isoparametric elements are taken to discretize the closed integral paths/crack surfaces and the outer boundary, respectively. The present method has distinct computational advantages in solving a fracture problem which has arbitrary numbers, distributions, orientations and shapes of cracks by a few boundary elements. Several examples are analysed and the computed results are in excellent agreement with other analytical or numerical solutions.     

Scaled boundary finite‐element analysis of a non‐homogeneous elastic half‐space

As a result of stresses experienced during and after the deposition phase, a soil strata of uniform material generally exhibits an increase in elastic stiffness with depth. The immediate settlement of foundations on deep soil deposits and the resultant stress state within the soil mass may be most accurately calculated if this increase in stiffness with depth is taken into account. This paper presents an axisymmetric formulation of the scaled boundary finite‐element method and incorporates non‐homogeneous elasticity into the method. The variation of Young's modulus (E) with depth (z) is assumed to take the form E=m_Ez^α, where m_E is a constant and αis the non‐homogeneity parameter. Results are presented and compared to analytical solutions for the settlement profiles of rigid and flexible circular footings on an elastic half‐space, under pure vertical load with αvarying between zero and one, and an example demonstrating the versatility and practicality of the method is also presented. Known analytical solutions are accurately represented and new insight regarding displacement fields in a non‐homogeneous elastic half‐space is gained. Copyright © 2003 John Wiley & Sons, Ltd.     

An implicit multi‐material Eulerian formulation

Multi‐material Eulerian methods were originally developed to solve problems in hypervelocity impact. They have proven to be useful for many other problems involving high‐strain rates such as the dynamic compaction of a powder and high‐speed machining. An implicit formulation has been developed to extend the range of applicability to quasi‐static problems such as hot isostatic pressing (HIP) and other material processing operations. Copyright © 2000 John Wiley & Sons, Ltd.     

An indirect elastodynamic boundary element method with analytic bases

An indirect time‐domain boundary element method (BEM) is presented here for the treatment of 2D elastodynamic problems. The approximated solution in this method is formulated as a linear combination of a set of particular solutions, which are called bases. The displacement and stress fields of a basis are analytically derived by means of solving Lame's displacement potentials. A semi‐collocation method is proposed to be the time‐stepping algorithm. This method is equivalent to a displacement discontinuity method with piecewise linear discontinuities in both space and time. The resulting time‐stepping scheme is explicit. The BEM is implemented to solve three numerical examples, Lamb's problem, half‐plane with a buried crack and Selberg's problem. Though Lamb's problem is considered a difficult problem for numerical methods, the current numerical results for the surface displacements show accurately the characteristics of the Rayleigh wave. This method is efficient and accurate. Copyright © 2003 John Wiley & Sons, Ltd.     

A unified 3D‐based technique for plate bending analysis using scaled boundary finite element method

This paper presents a unified technique for solving the plate bending problems by extending the scaled boundary finite element method. The formulation is based on the three‐dimensional governing equation without enforcing the kinematics of plate theory. Only the in‐plane dimensions are discretised into finite elements. Any two‐dimensional displacement‐based elements can be employed. The solution along the thickness is expressed analytically by using a matrix function. The proposed technique is consistent with the three‐dimensional theory and applicable to both thick and thin plates without exhibiting the numerical locking phenomenon. Moreover, the use of higher order spectral elements allows the proposed technique to better represent curved boundaries and to achieve high accuracy and fast convergence. Numerical examples of various plate structures with different thickness‐to‐length ratios demonstrate the applicability and accuracy of the proposed technique. Copyright © 2012 John Wiley & Sons, Ltd.     

Three‐dimensional steady thermal stress analysis by triple‐reciprocity boundary element method

The steady thermal stress problems without heat generation can be solved easily by the boundary element method. However, for the case with arbitrary heat generation, the domain integral is necessary. In this paper, it is shown that the problems of three‐dimensional steady thermal stress with heat generation can be approximately solved without the domain integral by the triple‐reciprocity boundary element method. In this method, an arbitrary distribution of heat generation is interpolated by boundary integral equations. In order to solve the problem, the values of heat generation at internal points and on the boundary are used. Copyright © 2005 John Wiley & Sons, Ltd.     

A new fast multipole boundary element method for solving large‐scale two‐dimensional elastostatic problems

A new fast multipole boundary element method (BEM) is presented in this paper for large‐scale analysis of two‐dimensional (2‐D) elastostatic problems based on the direct boundary integral equation (BIE) formulation. In this new formulation, the fundamental solution for 2‐D elasticity is written in a complex form using the two complex potential functions in 2‐D elasticity. In this way, the multipole and local expansions for 2‐D elasticity BIE are directly linked to those for 2‐D potential problems. Furthermore, their translations (moment to moment, moment to local, and local to local) turn out to be exactly the same as those in the 2‐D potential case. This formulation is thus very compact and more efficient than other fast multipole approaches for 2‐D elastostatic problems using Taylor series expansions of the fundamental solution in its original form. Several numerical examples are presented to study the accuracy and efficiency of the developed fast multipole BEM formulation and code. BEM models with more than one million equations have been solved successfully on a laptop computer. These results clearly demonstrate the potential of the developed fast multipole BEM for solving large‐scale 2‐D elastostatic problems. Copyright © 2005 John Wiley & Sons, Ltd.     

Singular‐ended spline interpolation for two‐dimensional boundary element analysis

A method of interpolation of the boundary variables that uses spline functions associated with singular elements is presented. This method can be used in boundary element method analysis of 2‐D problems that have points where the boundary variables present singular behaviour. Singular‐ended splines based on cubic splines and Overhauser splines are developed. The former provides C²‐continuity and the latter C¹‐continuity across element edges. The potentialities of the methodology are demonstrated analysing the dynamic response of a 2‐D rigid footing interacting with a half‐space. It is shown that, for a given number of elements at the soil–foundation interface, the singular‐ended spline interpolation increases substantially the displacement convergence rate and delivers smoother traction distributions. Copyright © 2000 John Wiley & Sons, Ltd.     

A fast multi‐level convolution boundary element method for transient diffusion problems

A new algorithm is developed to evaluate the time convolution integrals that are associated with boundary element methods (BEM) for transient diffusion. This approach, which is based upon the multi‐level multi‐integration concepts of Brandt and Lubrecht, provides a fast, accurate and memory efficient time domain method for this entire class of problems. Conventional BEM approaches result in operation counts of order O(N²) for the discrete time convolution over N time steps. Here we focus on the formulation for linear problems of transient heat diffusion and demonstrate reduced computational complexity to order O(N^3/2) for three two‐dimensional model problems using the multi‐level convolution BEM. Memory requirements are also significantly reduced, while maintaining the same level of accuracy as the conventional time domain BEM approach. Copyright © 2005 John Wiley & Sons, Ltd.