Dynamic elastoplastic analysis of 3-D structures by the domain/boundary element method

The main objective of this paper is to present a general three-dimensional boundary element methodology for solving transient dynamic elastoplastic problems. The elastostatic fundamental solution is used in writing the integral representation and this creates in addition to the surface integrals, volume integrals due to inertia and inelasticity. Thus, an interior discretization in addition to the usual surface discretization is necessary. Isoparametric linear quadrilateral elements are used for the surface discretization and isoparametric linear hexahedra for the interior discretization. Advanced numerical integration techniques for singular and nearly singular integrals are employed. Houbolt's step-by-step numerical time integration algorithm is used to provide the dynamic response. Numerical examples are presented to illustrate the method and demonstrate its accuracy.     

Analytical integral algorithm applied to boundary layer effect and thin body effect in BEM for anisotropic potential problems

A new completely analytical integral algorithm is proposed and applied to the evaluation of nearly singular integrals in boundary element method (BEM) for two-dimensional anisotropic potential problems. The boundary layer effect and thin body effect are dealt with. The completely analytical integral formulas are suitable for the linear and non-isoparametric quadratic elements. The present algorithm applies the analytical formulas to treat nearly singular integrals. The potentials and fluxes at the interior points very close to boundary are evaluated. The unknown potentials and fluxes at boundary nodes for thin body problems with the thickness-to-length ratios from 1E−1 to 1E−8 are accurately calculated by the present algorithm. Numerical examples on heat conduction demonstrate that the present algorithm can effectively handle nearly singular integrals occurring in boundary layer effect and thin body effect in BEM. Furthermore, the present linear BEM is especially accurate and efficient for the numerical analysis of thin body problems.     

Analysis of thin piezoelectric solids by the boundary element method

The piezoelectric boundary integral equation (BIE) formulation is applied to analyze thin piezoelectric solids, such as thin piezoelectric films and coatings, using the boundary element method (BEM). The nearly singular integrals existing in the piezoelectric BIE as applied to thin piezoelectric solids are addressed for the 2-D case. An efficient analytical method to deal with the nearly singular integrals in the piezoelectric BIE is developed to accurately compute these integrals in the piezoelectric BEM, no matter how close the source point is to the element of integration. Promising BEM results with only a small number of elements are obtained for thin films and coatings with the thickness-to-length ratio as small as 10⁻⁶, which is sufficient for modeling many thin piezoelectric films as used in smart materials and micro-electro-mechanical systems.     

A natural stress boundary integral equation for calculating the near boundary stress field

The stress computational accuracy of internal points by conventional boundary element method becomes more and more deteriorate as the points approach to the boundary due to the nearly singular integrals including nearly strong singularity and hyper-singularity. For calculating the boundary stress, a natural boundary integral equation in which the boundary variables are the displacements, tractions and natural boundary variables was established in the authors’ previous work. Herein, a natural stress boundary integral equation (NSBIE) is further proposed by introducing the natural variables to analyze the stress field of interior points. There are only nearly strong singular integrals in the NSBIE, i.e., the singularity is reduced by one order. The regularization algorithm proposed by the authors is taken over to deal with these singular integrals. Consequently, the NSBIE can analyze the stress field closer to the boundary. Numerical examples demonstrated that two orders of magnitude improvement in reducing the approaching degree can be achieved by NSBIE compared to the conventional one when the near boundary stress field is evaluated. Furthermore, this new way is extended to the multi-domain elasticity problem to calculate the stress field near the boundary and interface.     

Numerical evaluation of singular and finite-part integrals on curved surfaces using symbolic manipulation

The numerical integration of all singular surface integrals arising in 3-d boundary element methods is analyzed theoretically and computationally. For all weakly singular integrals arising in BEM, Duffy's triangular or local polar coordinates in conjunction with tensor product Gaussian quadrature are efficient and reliable for bothh-andp-boundary elements. Cauchy- and hypersingular surface integrals are reduced to weakly singular ones by analytic regularization which is done automatically by symbolic manipulation.     

Transient analysis of the dynamic stress intensity factors using SGBEM for frequency-domain elastodynamics

In this paper, a two-dimensional symmetric-Galerkin boundary integral formulation for elastodynamic fracture analysis in the frequency domain is described. The numerical implementation is carried out with quadratic elements, allowing the use of an improved quarter-point element for accurately determining frequency responses of the dynamic stress intensity factors (DSIFs). To deal with singular and hypersingular integrals, the formulation is decomposed into two parts: the first part is identical to that for elastostatics while the second part contains at most logarithmic singularities. The treatment of the elastostatic singular and hypersingular singular integrals employs an exterior limit to the boundary, while the weakly singular integrals in the second part are handled by Gauss quadrature. Time histories (transient responses) of the DSIFs can be obtained in a post-processing step by applying the standard fast Fourier transform (FFT) and algorithm to the frequency responses of these DSIFs. Several test examples are presented for the calculation of the DSIFs due to two types of impact loading: Heaviside step loading and blast loading. The results suggest that the combination of the symmetric-Galerkin boundary element method and standard FFT algorithms in determining transient responses of the DSIFs is a robust and effective technique.     

Evaluation of singular integrals in the symmetric Galerkin boundary element method

The implementation of the symmetric Galerkin boundary element method (SGBEM) involves extensive work on the evaluation of various integrals, ranging from regular integrals to hypersingular integrals. In this paper, the treatments of weak singular integrals in the time domain are reviewed, and analytical evaluations for the spatial double integrals which contain weak singular terms are derived. A special scheme on the allocation of Gaussian integration points for regular double integrals in the SGBEM is developed to improve the efficiency of the Gauss–Legendre rule. The proposed approach is implemented for the two-dimensional elastodynamic problems, and two numerical examples are presented to verify the accuracy of the numerical implementation.     

A variational approach to the contact angle dynamics of spreading droplets

The modeling and the numerical representation of the contact line between a two-phase interface and a solid surface are still open problems from the physical, mathematical and numerical point of view. This paper deals with the numerical simulation of the spreading of a single droplet impacting over horizontal dry surfaces. A new variational approach to study the droplet spreading is presented by coupling an interface front-tracking algorithm to the single-fluid finite element formulation of the incompressible Navier-Stokes equations which are solved on a fixed mesh. Standard no-slip boundary conditions near the contact line lead to a singular behavior that in the variational approach is removed by introducing a generalized boundary condition which is the sum of a dissipation term and the dynamical contact angle law. By changing the intensity of the dissipation a large number of boundary conditions around the contact point are modeled, ranging from no-slip to free-slip. Since the impact is over horizontal surfaces, axisymmetric solutions are investigated with high mesh resolutions. A very precise implementation of the capillary force with a volumetric extension of the curvature has been adopted. We have considered a Lagrangian front-tracking method to advect the interface. The marker representing the contact point is simply advected by the computed velocity at the boundary without the need to extrapolate the vector field from the interior and to enforce locally mass-conservation. The model has been tested for the impact and the spreading of a droplet on solid substrates with a different wettability at low Reynolds numbers where the inertial, the viscous and the surface tension forces are all important. A number of droplet impacts with different outcomes, ranging from simple deposition to partial and complete rebound, have been reproduced. However, our simulations indicate that the formulas suggested in the literature for the dynamical contact angle should be modified to simulate a broad class of experiments.     

An effective method for numerical evaluation of general 2D and 3D high order singular boundary integrals

In this paper, a robust method is presented for numerical evaluation of weakly, strongly, hyper- and super-singular boundary integrals, which exist in the Cauchy principal value sense in two- and three-dimensional problems. In this method, the singularities involved in integration kernels are analytically removed by expressing the non-singular parts of the integration kernels as power series in the local distance ρ of the intrinsic coordinate system. For three-dimensional boundary integrals, the radial integration method [1] is applied to transform the surface integral into a line integral over the contour of the surface and to remove various orders of singularities within the radial integrals. Some examples are provided to verify the correctness and robustness of the presented method.     

3D BEM for the general piezoelectric solids

In this paper we present a new scheme of three-dimensional boundary element method for the general anisotropic piezoelectric solids. We use the Radon transform representation of the three-dimensional fundamental solutions of piezoelectricity and integrate them analytically over the triangular boundary element with the linear interpolation. This reduces the computation for the system matrices G and H from the standard singular surface integrations to the simple regular line integrations and enables a drastic reduction of the computation time. The integrand of the line integral consists of the product of a function dependent and another function independent on the location vectors representing the source and observation points. The latter function depends only on the material and element properties and thus calculated only once for each element and saved for a repeated use in the calculation of G and H matrices and in the post-processing. Exploitation of this favorable structure results in the further reduction of the computation time for very large systems. The implementation of the proposed method with numerical examples will be presented.     

Analysis of three-dimensional anisotropic heat conduction problems on thin domains using an advanced boundary element method

In this paper, an advanced boundary element method (BEM) is developed for solving three-dimensional (3D) anisotropic heat conduction problems in thin-walled structures. The troublesome nearly singular integrals, which are crucial in the applications of the BEM to thin structures, are calculated efficiently by using a nonlinear coordinate transformation method. For the test problems studied, promising BEM results with only a small number of boundary elements have been obtained when the thickness of the structure is in the orders of micro-scales (10^?6), which is sufficient for modeling most thin-walled structures as used in, for example, smart materials and thin layered coating systems. The advantages, disadvantages as well as potential applications of the proposed method, as compared with the finite element method (FEM), are also discussed.     

A node-centric indirect boundary element method: three-dimensional displacement discontinuities

An indirect boundary element formulation based on unknown physical values, defined only at the nodes (vertices) of a boundary discretization of a linear elastic continuum, is introduced. As an adaptation of this general framework, a linear displacement discontinuity density distribution using a flat triangular boundary discretization is considered. A unified element integration methodology based on the continuation principle is introduced to handle regular as well as near-singular and singular integrals. The boundary functions that form the basis of the integration methodology are derived and tabulated in the appendix for linear displacement discontinuity densities. The integration of the boundary functions is performed numerically using an adaptive algorithm which ensures a specified numerical accuracy. The applications include verification examples which have closed-form analytical solutions as well as practical problems arising in rock engineering. The node-centric displacement discontinuity method is shown to be numerically efficient and robust for such problems.     

Bending of anisotropic plates by charge simulation method

A boundary element method, called the charge simulation method, is presented for analysis of anisotropic thin-plate bending problems. In this method the singular integrals involved in the other boundary element methods are eliminated and there is no numerical integration involved. Further, the domain integral is replaced by a polynomial particular integral; hence the domain discretization is avoided. This method is conceptually very simple. The results obtained by this method are compared with the available analytical solutions for various anisotropic and symmetric laminates and the results are in good agreement.     

Recent developments in the Trefftz method for finite element and boundary element applications

In 1926 E. Trefftz published a paper about a variational formulation which utilizes boundary integrals. Almost half a century later researchers became interested again in the ideas of Trefftz when the potential advantage of the Trefftz-method for an efficient use in numerical application on a computer was recognized. The concept of Trefftz can be used both for finite element and boundary element applications. A crucial ingredient of the Trefftz- method is a set of linearly independent trial functions which a priori satisfy the governing differential equations under consideration. In this paper an overview of some recent developments to construct trial functions for the Trefftz-method in a systematic manner is given. Using different types of approximation functions (singular or non-singular) we can obtain very accurate finite element and boundary element algorithms.     

A method for the computation of inertial properties for general areas

A method is presented to compute the inertial properties for shapes whose boundary is not expressible in terms of simple mathematical functions. The boundary is approximated in terms of local coordinates. The relevant double integrals are transformed into line integrals and then evaluated. A numerical example is presented to demonstrate the versatility of the algorithm.     

Boundary element analysis of nonlinear transient heat conduction problems

This paper is concerned with a boundary element formulation and its numerical implementation for the nonlinear transient heat conduction problems with temperature-dependent material properties. By using the Kirchhoff transformation for the material properties a set of pseudo-linear integral equations is obtained in space and time for the fully three-dimensional nonlinear problems under consideration. The resulting boundary integral equations are solved by means of the usual boundary element method. Emphasis is placed on the numerical solution procedure employing constant elements with respect to time. It is shown that in this case there is no need to evaluate the domain integrals resulting from the nonlinearity of the problem. Finally, the powerful usefulness of the proposed method is demonstrated through the numerical computation of several sample problems.     

Quadrature methods for periodic singular and weakly singular Fredholm integral equations

High-accuracy numerical quadrature methods for integrals of singular periodic functions are proposed. These methods are based on the appropriate Euler-Maclaurin expansions of trapezoidal rule approximations and their extrapolations. They are subsequently used to obtain accurate quadrature methods for the solution of singular and weakly singular Fredholm integral equations. Throughout the development the periodic nature of the problem plays a crucial role. Such periodic equations are used in the solution of planar elliptic boundary value problems such as those that arise in elasticity, potential theory, conformal mapping, free surface flows, etc. The use of the quadrature methods is demonstrated with numerical examples.     

A nonlinear,semi-analytical finite element analysis for nearly axisymmetric solids

The presented research is concerned with the development of the theory and accompanying computer program for a semi-analytical finite element analysis of non axisymmetrically loaded, nearly axisymmetric solids. The theoretical basis of the method together with numerical procedures for handling boundary conditions, rigid body motion and the iterative solution process are described. A finite element program for evaluating the analysis is discussed, evaluated for effectiveness and applied to several examples.The range of application of the analysis for inhomogeneous, orthotropic, nonlinear and nearly axisymmetric bodies is demonstrated by a series of examples. The substantial savings in computer time and memory as compared to a conventional finite element analysis is discussed.     

A finite element based level-set method for multiphase flow applications

A numerical method for simulating incompressible two-dimensional multiphase flow is presented. The method is based on a level-set formulation discretized by a finite-element technique. The treatment of the specific features of this problem, such as surface tension forces acting at the interfaces separating two immiscible fluids, as well as the density and viscosity jumps that in general occur across such interfaces, have been integrated into the finite-element framework. Using a method based on the weak formulation of the Navier-Stokes equations has its advantages. In this formulation, the singular surface tension forces are included through line integrals along the interfaces, which are easily approximated quantities. In addition, differentiation of the discontinuous viscosity is avoided. The discontinuous density and viscosity are included in the finite element integrals. A strategy for the evaluation of integrals with discontinuous integrands has been developed based on a rigorous analysis of the errors associated with the evaluation of such integrals. Numerical tests have been performed. For the case of a rising buoyant bubble the results are in good agreement with results from a front-tracking method. The run presented here is a run including topology changes, where initially separated areas of one fluid merge in different stages due to buoyancy effects. Received: 1 March 1999 / Accepted: 17 June 1999     

A 2.5D coupled FE–BE methodology for the dynamic interaction between longitudinally invariant structures and a layered halfspace

This paper presents a general 2.5D coupled finite element–boundary element methodology for the computation of the dynamic interaction between a layered soil and structures with a longitudinally invariant geometry, such as railway tracks, roads, tunnels, dams, and pipelines. The classical 2.5D finite element method is combined with a novel 2.5D boundary element method. A regularized 2.5D boundary integral equation is derived that avoids the evaluation of singular traction integrals. The 2.5D Green’s functions of a layered halfspace, computed with the direct stiffness method, are used in a boundary element method formulation. This avoids meshing of the free surface and the layer interfaces with boundary elements and effectively reduces the computational efforts and storage requirements. The proposed technique is applied to four examples: a road on the surface of a halfspace, a tunnel embedded in a layered halfspace, a dike on a halfspace and a vibration isolating screen in the soil.