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Summary In order to investigate size-dependent creeping plane microfluidic flow, a boundary element method is implemented that involves the calculation of interior quantities and multi-domain problems. The governing equations are formulated using the skew-symmetric character of the couple-stress tensor and its energy conjugate mean-curvature tensor, as established in consistent couple stress theory. This theoretical formulation includes one characteristic material length scale parameter l that represents the size-dependency of the problem. Here, we present the boundary integral representation and numerical implementation for two-dimensional size-dependent steady state creeping incompressible flow, in which velocities, angular velocities, force- and couple-tractions are the primary variables. Beyond the previously known singular fundamental solution kernels for point force and point couple in the velocity and angular velocity integral equations, here we present the integral equations for calculating internal mean curvatures and couple-stresses, along with their corresponding kernels. The formulation is then applied to solve some computational problems both to investigate the size effects on the response resulting from the theory, and to validate the strength of the numerical method. For this purpose, we study the important effect of different boundary conditions on the flow pattern and consider a multi-fluid domain Couette flow problem.

A boundary element method is developed to examine two-dimensional size-dependent thermoelastic response in isotropic solids. The formulation is founded on the recently established consistent couple stress theory, in which both the couple-stress tensor and its energy conjugate mean curvature tensor are skew-symmetric. For isotropic materials, there is no thermal mean curvature deformation, and the thermoelastic effect is solely the result of thermal strain deformation. As a result, size-dependency is quantified by one characteristic material length scale parameter l , while the thermal coupling is activated through the classical thermal expansion coefficient α . Interestingly, in this size-dependent multi-physics model, the thermal governing equation is independent of the deformation. However, the mechanical governing equations depend on the temperature field. Here, we develop the boundary integral representation and numerical implementation for this size-dependent thermoelastic boundary element method (BEM) for plane problems, which involves temperatures, displacements, rotations, normal heat fluxes, force-tractions and couple-tractions as primary variables within a boundary-only formulation. Then, we apply this new BEM formulation to several basic computational problems in an effort to validate the robustness of the numerical implementation and to examine size-dependent response.

Computational mechanics codes include contact or over-closure models to account for contact stiffness, which can be related to surface roughness and nominal pressure from Greenwood and Williamson. We present results of wear simulation studies with adjustable contact parameters for a ring-on-ring sliding configuration. A boundary element code is used to solve the non-linear contact problem with an axisymmetric thermoelastic representation of the rings, along with a localized Archard wear model. Parametric studies indicate the extent to which the wear tracks can be affected by the roughness and wear history. The effects range from minor to moderately significant, with areas of maximum wear shifting radially. Contact parameters influence pressure, temperature and wear distributions, with stiffer contacts resulting in greater localization.

Within the framework of continuum fluid dynamics, couple-stresses appear as an inevitable consequence of non-central forces and the discrete character of matter at the finest scales. As a result, the force-stress tensor becomes non-symmetric and classical theory may not accurately predict the behavior. Recent theoretical work has shown that the couple-stress tensor must be skew-symmetric and that mean curvature rate is the energy-conjugate kinematical measure. The resulting fully consistent couple stress theory incorporates a characteristic material length scale for the fluid that becomes increasingly important, as the characteristic geometric dimension of the problem approaches that level. This size-dependent non-Newtonian theory is essential to understand a range of behavior at micro-scales with potential applications to blood flow and lubrication, among others. Here we concentrate on steady creeping flow within this newly developed fully determinate linear couple stress theory and formulate a boundary integral representation for two-dimensional problems. These boundary integral equations are written in terms of velocities, angular velocities, force-tractions and moment-tractions as primary variables. Details are provided on the derivation of fundamental solutions and on the corresponding boundary element implementation. Afterwards, the new boundary element method is applied for the solution of two basic problems to explore some consequences of this size-dependent couple stress fluid mechanics.

SUMMARY Couple-stresses are introduced to account for the microstructure of a material within the framework of continuum mechanics. Linear isotropic versions of such materials possess a characteristic material length l that becomes increasingly important as problem dimensions shrink to that level (e.g., as the radius a of a critical hole reduces to a size comparable to l). Consequently, this size-dependent elastic theory is essential to understand the behavior at micro- and nano-scales and to bridge the atomistic and classical continuum theories. Here we develop an integral representation for two-dimensional boundary value problems in the newly established fully determinate theory of isotropic couple stress elastic media. The resulting boundary-only formulation involves displacements, rotations, force-tractions and moment-tractions as primary variables. Details on the corresponding numerical implementation within a boundary element method are then provided, with emphasis on kernel singularities and numerical quadrature. Afterwards the new formulation is applied to several computational examples to validate the approach and to explore the consequences of size-dependent couple stress elasticity. Copyright © 2011 John Wiley & Sons, Ltd.

A new boundary element formulation is developed to analyze two-dimensional size-dependent thermoelastic response in linear isotropic couple stress materials. The model is based on the recently developed consistent couple stress theory, in which the couple-stress tensor is skew-symmetric. The size-dependency effect is specified by one characteristic parameter length scale l, while the thermal effect is quantified by the classical thermal expansion coefficient α and conductivity k. We discuss the boundary integral formulation and numerical implementation of this size-dependent thermoelasticity boundary element method (BEM). Then, we apply the resulting BEM formulation to a computational example to validate the numerical implementation and to explore thermoelastic couplings as the non-dimensional characteristic scale of the problem is varied. Interestingly, for a cantilever beam with a transverse temperature gradient, we find significantly reduced non-dimensional tip deflections as the beam depth h approaches the material characteristic length scale l. On the other hand, when l/h < 0.01, the classical size-independent deflections are recovered.Copyright © 2015 by ASME

Recently, we have developed multi-level boundary element methods (MLBEM) for the solution of the Laplace and Helmholtz equations that involve asymptotically decaying non-oscillatory and oscillatory singular kernels, respectively. The accuracy and efficiency of the fast boundary element methods for steady-state heat diffusion and acoustics problems have been investigated for square domains. The current work extends the MLBEM methodology to the solution of Stokes equation in more complex two-dimensional domains. The performance of the fast boundary element method for the Stokes flows is first investigated for a model problem in a unit square. Then, we consider an example problem possessing an analytical solution in a rectangular domain with 5:1 aspect ratio, and finally, we study the performance of the MLBEM algorithm in a C-shaped domain.

Several recently developed boundary element formulations for time-dependent convective heat diffusion appear to provide very efficient computational tools for transient linear heat flows. More importantly, these new approaches hold much promise for the numerical solution of related nonlinear problems, e.g., Navier–Stokes flows. However, the robustness of these methods has not been examined, particularly for high Peclet number regimes. Here, we focus on these regimes for two-dimensional problems and develop the necessary temporal and spatial integration strategies. The algorithm takes advantage of the nature of the time-dependent convective kernels, and combines analytic integration over the singular portion of the time interval with numerical integration over the remaining nonsingular portion. Furthermore, the character of the kernels lets us define an influence domain and then localize the surface and volume integrations only within this domain. We show that the localization of the convective kernels becomes more prominent as the Peclet number of the flow increases. This leads to increasing sparsity and in most cases improved conditioning of the global matrix. Thus, iterative solvers become the primary choice. We consider two representative example problems of heat propagation, and perform numerical investigations of the accuracy and stability of the proposed higher-order boundary element formulations for Peclet numbers up to 105.

A higher-order boundary element method recently developed by the current authors [Numer. Heat Trans. Part B: Fundamentals 45 (2004) 109] for two-dimensional steady convective heat diffusion is generalized to three-dimensions. In order to facilitate an accurate and efficient boundary element formulation, we introduce an influence domain due to these convective kernels and then localize the surface integrations only within the domain of influence. The localization of the kernels becomes more prominent as the Peclet number of the flow increases. This, in turn, leads to increasing sparsity and improved conditioning of the global matrix. Consequently, iterative solvers for sparse matrices become the primary choice. In this paper, we consider an example problem with an exact solution and investigate the accuracy and efficiency of the boundary element formulation for Peclet numbers in the range from 10 to 1000. The bi-quartic boundary elements included in this study are shown to provide acceptable levels of resolution.

Most recently, we have developed a novel multilevel boundary element method (MLBEM) for steady Stokes flows in irregular two-dimensional domains (Grigoriev, M.M., and Dargush, G.F., Comput. Methods. Appl. Mech. Eng., 2005). The multilevel algorithm permitted boundary element solutions with slightly over 16,000 degrees of freedom, for which approximately 40-fold speedups were demonstrated for the fast MLBEM algorithm compared to a conventional Gauss elimination approach. Meanwhile, the sevenfold memory savings were attained for the fast algorithm. This paper extends the MLBEM methodology to dramatically improve the performance of the original multilevel formulation for the steady Stokes flows. For a model problem in an irregular pentagon, we demonstrate that the new MLBEM formulation reduces the CPU times by a factor of nearly 700,000. Meanwhile, the memory requirements are reduced more than 16,000 times. These superior run-time and memory reductions compared to regular boundary element methods are achieved while preserving the accuracy of the boundary element solution.

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(N 2 ) 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.

A higher-order boundary element method (BEM) recently developed by the current authors (Comput Methods Appl Mech Eng 2003; 192: 4281–4298; 4299–4312; 4313–4335) for time-dependent convective heat diffusion in two-dimensions appears to be a very attractive tool for efficient simulations of transient linear flows. However, the previous BEM formulation is restricted to relatively small time step sizes (i.e. Δt⩽4κ/V2) owing to the convergence issues of the time series for the kernel representation within a time interval. This paper extends the boundary element formulation in a way to allow time step sizes several orders of magnitude larger than in the previous approach. We consider an example problem of thermal propagation, and investigate the accuracy and efficiency of BEM formulations for Peclet numbers in the range from 103 to 105. Copyright © 2005 John Wiley & Sons, Ltd.

Higher-order boundary element methods (BEM) are presented for three-dimenisonal steady convective heat diffusion at high Peclet numbers. The boundary element formulation is facilitated by the definition of an influence domain due to convective kernels. This approach essentially localizes the surface integrations only within the domain of influence which becomes more narrowly focused as the Peclet number increases. The outcome of this phenomenon is an increased sparsity and improved conditioning of the global matrix. Therefore, iterative solvers for sparse matrices become a very efficient and robust tool for the corresponding boundary element matrices. In this paper, we consider an example problem with an exact solution and investigate the accuracy and efficiency of the higher-order BEM formulations for high Peclet numbers in the range from 1000 to 100,000. The bi-quartic boundary elements included in this study are shown to provide very efficient and extremely accurate solutions to this problem, even on a single engineering workstation.

A fast, accurate, and efficient multilevel boundary-element method (MLBEM) is developed to solve general boundary-value problems arising in computational mechanics. Here we concentrate on problems of two-dimensional steady potential flow and present a fast, direct boundary-element formulation. This novel method extends the pioneering work of Brandt and Lubrecht on multilevel multi-integration (MLMI) in several important ways to address problems with mixed boundary conditions. We utilize bi-conjugate gradient methods (BCGMs) and implement the MLMI approach for fast matrix and matrix transpose multiplication for every iteration loop. After introducing a C-cycle multigrid algorithm, we find that the number of iterations for the bi-conjugate gradient methods is independent of the boundary-element mesh discretization for a broad range of steady-state heat diffusion problems. Here, for a model problem in an L-shaped domain, we demonstrate that the computational complexity of the proposed method approaches the d...

Higher-order boundary-element methods (BEM) are presented for steady-state convective diffusion problems in two dimensions. The free-space steady convective diffusion fundamental solutions considered in this article provide an analytical upwinding for the entire Peclet number range, from zero to infinity. However, integration of the kernels over the boundary elements requires considerable attention, especially at higher Peclet numbers. We define an influence domain due to these convective kernels and then localize the surface integrations only within the domain of influence. The localization of the kernels becomes more prominent as the Peclet number of the flow increases. This, in turn, leads to increasing sparsity and improved conditioning of the global matrix. Consequently, iterative solvers become the primary choice. We consider an example problem with an exact solution, and investigate the accuracy and efficiency of the higher-order BEM formulations for Peclet numbers in the range from 200 to 200,000....

Most recently, we have developed a novel multi-level boundary element method (MLBEM) for the solution of the steady heat diffusion equation involving asymptotically decaying non-oscillatory log-singular and strongly singular kernels. This hierarchical approach generalizes the pioneering work of Brandt and Lubrecht on multi-level multi-integration (MLMI) and C-cycle multi-grid to the broader class of mixed boundary value problems. The result is a fast, accurate and efficient boundary element method. The present paper extends this new computational methodology to the solution of the Helmholtz equation involving oscillatory log-singular and strongly singular kernels for two-dimensional problems. We consider a direct boundary element formulation and, due to the nature of the fundamental solutions, split the corresponding boundary integral equation into real and imaginary parts. Then, we introduce double-noded corners to facilitate a patch-by-patch application of the MLMI algorithm for fast matrix–vector and matrix-transpose–vector multiplications within bi-conjugate gradient methods. The performance of the proposed fast MLBEM is investigated using a numerical example that possesses an exact solution. For wave numbers κ=20 and below, we demonstrate that the fast MLBEM algorithm for the Helmholtz equation is robust, accurate, and exceptionally efficient.

A boundary element method is developed for the determination of generalized stress intensity factors associated with bi-material interfaces. The new formulation, which is based on the theory of boundary eigensolutions, employs displacements and weighted tractions as primary variables. As a result, the solution is obtained exclusively in terms of bounded quantities enabling the attainment of mesh-independent results while requiring only surface discretization. Several numerical examples are considered to assess the performance of the method, including a detailed strength analysis of epoxy-steel cylindrical butt joints.

A boundary element method for transient convective diffusion phenomena presented in Part I of the paper is extended to two dimensional problems. We introduce a series representation for the transient convective kernel and perform a time integration for the double integrals to evaluate coefficients of the time-discrete boundary integral equation. The time-integrated kernels are evaluated for the linear, quadratic and quartic time interpolation functions utilized in the paper. Then, linear, quadratic and quartic boundary elements as well as bi-linear, bi-quadratic and bi-quartic volume cells are introduced to ensure proper resolution in space for the two-dimensional formulation. Due to the singular nature of the transient convective diffusion kernels, integration of the kernels over the boundary elements and volume cells requires a considerable effort to maintain a desired level of accuracy. We define influence domains due to time-integrated and time-delayed kernels arising for the surface and volume integrals, respectively. Note that the kernel influences are extremely localized due to the convective nature of the kernels, thus, the surface and volume integrations are performed only within these domains of influence. The localization of the kernels becomes more prominent as the Peclet number of the flow increases. Due to increasing sparsity of the global matrix, iterative solvers become the primary choice for the convective diffusion problems.

Higher-order boundary element methods (BEM) for transient convective diffusion phenomena presented in Parts I and II of the paper are implemented numerically to examine their performance for a series of model problems. In order to highlight the importance of proper resolution in time and space for the transient problems, we consider both the one- and two-dimensional formulations in this paper. We utilize an implicit time-recurring formulation to advance in the time domain. The use of higher-order time interpolation functions facilitates extremely high resolution with respect to time step sizes, and thus necessitates similar levels of spatial resolution. In this paper, linear, quadratic and quartic boundary elements as well as bi-linear, bi-quadratic and bi-quartic volume cells are implemented to ensure a desired high level of accuracy both in time and in space. In order to investigate the performance of the BEM formulations, we introduce five problems of unsteady convection–diffusion that possess exact solutions. For all five numerical examples considered in this paper, the higher-order BEM demonstrate an extremely high level of accuracy even for predominantly convective flows. It is shown that the use of the convective kernels provide an analytical upwinding for any Peclet number and any mesh orientation with respect to the flow direction. The introduction of the kernel influence domains facilitates a very efficient and robust algorithm for integration over boundary elements and volume cells. Due to the nature of the convective kernels, the resulting global matrix is sparse with the non-zero elements localized around the main diagonal. The preconditioned bi-conjugate gradient method utilized in the paper allows very efficient factorization of the global sparse matrix. Moreover, the efficiency of the BEM formulations increases dramatically with the increase of the Peclet number of the flow.

A general formulation of higher-order boundary element methods (BEM) is presented for time-dependent convective diffusion problems in one- and multi-dimensions. Free-space time-dependent convective diffusion fundamental solutions originally proposed by Carslaw and Jaeger are used to obtain the boundary integral formulation. Linear, quadratic and quartic time interpolation functions are introduced in this paper for approximate representation of time-dependent boundary temperatures and normal fluxes. Closed form time integration of the kernels is mandatory to attain both accuracy and efficiency of the numerical approach. A complete set of time integrals for the one-dimensional formulation is presented here for the first time in the literature.

The theory of fundamental boundary eigensolutions for elastostatic problems, developed in Part I, is applied to formulate methods for computational mechanics. This theory shows that every elastic solution can be written as a linear combination of some fundamental boundary orthogonal deformations, thus providing a generalized Fourier expansion. One finds that traditional boundary element and finite element methods are largely consistent with this theory, but do not harness its full power. This theory shows that these computational methods are indirectly a generalized discrete Fourier analysis. Furthermore, by utilizing suitable boundary weight functions, boundary element and finite element formulations may be written exclusively in terms of bounded quantities, even for non-smooth problems involving notches, cracks, mixed boundary conditions and bi-material interfaces. The close relationship between the resulting boundary element and finite element methods also becomes evident. Both use displacement and surface traction as primary variables. A new degree-of-freedom concept is introduced, along with a stiffness tensor that enables one to visualize a finite element method via a boundary discretization process, just as in a boundary element approach. Global convergence characteristics of the traction-oriented finite element method are also developed. Comparisons with closed-form fundamental boundary eigensolutions for a circular elastic disc are presented in order to provide a means for assessing the numerical methods. Several other numerical examples are solved efficiently by using the concept of boundary eigensolutions in an indirect fashion. The results indicate that the algorithms follow the underlying theory and that solutions to non-smooth problems can be obtained in a systematic manner. Beyond this, the concept of boundary eigensolutions provides an alternative view of computational continuum mechanics that may lead to the development of other non-traditional approaches.

Despite the significant number of publications on boundary element methods (BEM) for time-dependent problems of heat diffusion, there still remain issues that need to be addressed, most importantly accuracy of the numerical modelling. Although very precise for steady-state problems, the common boundary element methods applied to transient problems do not yield highly accurate numerical solutions. This paper investigates the reasons that prohibit achievement of a high level of accuracy for transient heat diffusion problems with continuous temperature and bounded heat flux solutions. In order to greatly enhance the commonly used boundary element formulations, we propose higher-order time interpolation functions, including quadratic and quartic approximations. We show that the use of higher-order time functions greatly reduces the numerical error concentrated in the corner regions, and results in very good uniformity of the flux and temperature distributions along the boundaries for problems where uniform distributions are expected. In order to highlight the importance of proper resolution both in time and space for the transient problems, we consider one- and two-dimensional formulations in this paper. High-order boundary elements using quartic shape functions, as well as high-order bi-quartic volume cells, are used to attain mesh-independent numerical solutions. We consider four transient heat diffusion problems that possess exact solutions to investigate the convergence rate and accuracy of the higher-order boundary element formulations. A very high level of accuracy is possible for both one- and two-dimensional formulations. Additionally, we show that the accuracy of a commercially available finite-element code is far less than that of the boundary element methods for a given spatial and temporal discretization. Copyright © 2002 John Wiley & Sons, Ltd.

A theory of boundary eigensolutions is presented for boundary value problems in engineering mechanics. While the theory is quite general, the presentation here is restricted to potential problems. Contrary to the traditional approach, the eigenproblem is formed by inserting the eigenparameter, along with a positive weight function, into the boundary condition. The resulting spectra are real and the eigenfunctions are mutually orthogonal on the boundary, thus providing a basis for solutions. The weight function permits effective treatment of nonsmooth problems associated with cracks, notches and mixed boundary conditions. Several ideas related to the convergence characteristics are also introduced. Furthermore, the connection is made to integral equation methods and variational methods. This paves the way toward the development of new computational formulations for finite element and boundary element methods. Two numerical examples are included to illustrate the applicability.

A boundary element method (BEM) for steady viscous fluid flow at high Reynolds numbers is presented. The new integral formulation with a poly-region approach involves the use of the convective kernel with slight compressibility that was previously employed by Grigoriev and Fafurin [1] for driven cavity flows with Reynolds numbers up to 1000. In order to avoid the overdeterminancy of the global set of equations when using eight-noded rectangular volume cells from that previous work, 12-noded hexagonal volume regions are introduced. As a result, the number of linearly independent integral equations for each node becomes equal to the degrees of freedom of the node. The numerical results for square-driven cavity flow having Reynolds numbers up to 5000 are compared to those obtained by Ghia et al. [2] and demonstrate a high level of accuracy even in resolving the secondary vortices at the corners of the cavity. Next, a comprehensive study is done for backward-facing step flows at Re=500 and 800 using the BEM, along with a standard Galerkin-based finite element methods (FEM). The numerical methods are in excellent agreement with the benchmark solution published by Gartling [3]. However, several additional aspects of the problem are also considered, including the effect of the inflow boundary location and the traction singularity at the step corner. Furthermore, a preliminary comparative study of the poly-region BEM versus the standard FEM indicates that the new method is more than competitive in terms of accuracy and efficiency. Copyright © 1999 John Wiley & Sons, Ltd.

Objective The aim of this study is to explain the influence of peripheral interface stress singularities on the testing of tensile bond strength. The relationships between these theoretically predicted singularities and the effect of specimen size on the measured bond strength are evaluated. Methods Finite element method (FEM) and boundary element method (BEM) analyses of microtensile bond strength test specimens were performed and the presence of localized high stress concentrations and singularities was analyzed. The specimen size effect predicted by the models was compared to previously published experimental data. Results FEM analysis of single-material trimmed hour-glass versus cast cylindrical specimens showed different theoretical stress distributions, with the dumbbell or cylindrical specimens showing a more homogeneous distribution of the stress on the critical symmetry plane. For multi-material specimens, mathematical singularities at the free edge of the bonded interface posed a computational challenge that resulted in mesh-dependence in the standard FEM analysis. A specialized weighted-traction BEM analysis, designed to eliminate mesh-dependence by capturing the effect of the singularity, predicted a specimen size effect that corresponds to that published previously in the literature. Significance The results presented here further support the attention to specimen dimensions that has already broadened the empirical use of the microtensile test methods. FEM and BEM analyses that identify stress concentrations and especially marginal stress singularities must be accounted for in reliable bonding strength assessments. Size-dependent strength variations generally attributed to the effects of flaw distributions throughout the interfacial region are not as relevant as the presence of singularities at bonded joint boundaries – as revealed by both FEM and BEM analyses, when interpreted from a generalized fracture mechanics perspective. Furthermore, this size-dependence must be considered when evaluating or designing dental adhesive systems.

A boundary element method (BEM) is presented for the dynamic analysis of axisymmetric poroelastic media, subjected to time-harmonic excitation. The integral formulation, based on the fundamental solution of Biot's poroelastic theory, eliminates the need for domain discretization for piecewise homogeneous media and automatically satisfies the radiation conditions. Additionally, there is no special treatment required for the incompressibilty that may result under undrained conditions. The axisymmetric formulation that is developed offers further reduction in dimensionality of the problem and provides a convenient framework for investigating the behavior of circular foundations. The BEM algorithm is validated via comparisons with an analytical solution for the dynamic response of a soil column under uniaxial loading and is then utilized to study the dynamic compliance of a shallow circular foundation. The suitability of employing various quasistatic and elastodynamic idealizations is considered in some detail.

Thermal expansion of the part can be a significant source of dimensional and form errors in precision machining operations. This paper describes a method for calculating the thermal expansion of an axisymmetric workpiece. The analysis is based on a commercially available boundary element code modified to properly represent concentrated moving heat sources such as those produced in machining. The inputs required are the amount of heat entering the part from the cutting zone and the thermal properties of the workpiece material. Calculations are compared with direct measurements of expansion from tests on large diameter 2024 aluminum sleeves. The agreement between calculated and measured values is generally reasonable, although calculated expansions are consistently smaller than measured expansions. This error is probably due to errors in estimating the heat input to the part, and particularly the neglect of flank friction in heat input calculations. Sample calculations for hard turning of a wheel spindle show that expansions can approach tolerances on critical surfaces. Based on sample calculations, thermal expansion is likely to be significant when hard turning parts with tolerances on the order of 0.01 mm. For these applications, critical surfaces should be machined first, before cuts on other sections heat the part, and gaging should be carried out only after the part has cooled.

A solution methodology is presented for computing the dynamic stiffnesses in all vibration modes of viscoelastic-fluid dampers with mechanical properties that depend strongly on both frequency and temperature. The method of reduced variables is introduced to construct the master curves of the dynamic modulus of the fluid at some reference temperature from experimental data at different temperatures. Three generalized-derivative (fractional and complex order) constitutive models are proposed to approximate the dynamic modulus of the polybutane fluid over a wide frequency range. The complex parameters of the proposed models are obtained by finding the best fitting of the master curves. The solution of the problem is obtained by transforming the constitutive and balance equations in the Laplace domain, and developing a boundary-element formulation. The resulting method is applied in the prediction of the mechanical properties of a damper containing the polybutane viscoelastic fluid. Results using the three p...

A boundary element method is developed for transient and time harmonic analysis of problems in dynamic poroelasticity and generalized thermoelasticity, involving both two- and three-dimensional geometries. Laplace domain infinite space fundamental solutions are employed to produce a formulation that requires only surface discretization. Consequently, the resulting algorithm provides an attractive alternative to existing volume-based methods, particularly for media of infinite extent. Details of the formulation and numerical implementation are presented. Several applications are included to validate the method and to emphasize certain aspects of the dynamic theory.

A time domain boundary element method (BEM) for evaluating stresses in an axisymmetric soil mass undergoing consolidation has been developed. Previous BEM work on axisymmetric poroelasticity for boundary displacements and pore pressures is extended to permit the computation of stresses at both boundary and interior points. The stress formulation preserves the surface-only discretization. The boundary displacement integral equation is progressively differentiated to obtain the related stress and strain integral equations. Explicit expressions for the steady-state axisymmetric fundamental solutions are derived in this process. The transient components of the integrands are obtained directly from the transformation of the three-dimensional kernels into a cylindrical system. Numerical implementation of these integral equations is carried out within a general purpose BEM computer code and several illustrative examples are presented to validate the method.

A boundary element method (BEM) for general thermoplastic analysis is presented. A thermally-sensitive two surface material model is used to simulate constitutive behavior. The nonlinear system of equations, arising from an initial stress approach, is solved using a new Newton-Raphson solution algorithm. The stress tensors are collapsed into a scalar variable using certain characteristic tensors arising out of the constitutive equations. This leads to a simpler system requiring much less computation. The solution algorithm is iterative in nature and checks for convergence at the end of each increment. The resulting robust analysis tool is then utilized in solving several practical applications to illustrate the growing importance of BEM as an alternative to the finite element method.

A general boundary-element formulation is presented for the predic- tion of the dynamic response of fluids with viscoelastic behavior. The fluid is modeled by a generalized constitutive relation that contains either complex-valued parameters and complex-order derivatives or real-valued parameters and fractional- order derivatives, These models are consistent with basic theories and are not arbitrary constructions. The models are valid for linear viscoelastic fluid behavior and are limited to fluid motions with infinitesimally small displacement gradients. The governing equations are transformed into the Laplace domain and the infinite space fundamental solution is derived. The resulting integral equations are then solved by numerical procedures. The method is applied in the prediction of the dynamic mechanical properties of a viscous damper containing a viscoelastic fluid in the form of silicon gel. The fluid is modeled by a fractional derivative Maxwell model. The predicted mechanical properties of the device are found to be in ex- cellent agreement with experimental results.

The stress intensity factors for cracks in three-dimensional, thermally stressed structures are computed by using the boundary element method. While many boundary and volume-integral-based formulations are available for the treatment of thermoelastic problems in solids, the present analysis is based on a recently developed boundary-only formulation. The accuracy of the solutions in the present work is improved by using special elements at the crack front that accurately model the variation of displacement, temperature fields and singularity of traction, flux fields.

A time dependent boundary element method (BEM) for evaluating thermal stresses in an axisymmetric thermoelastic body has been developed. Previous BEM work on axisymmetric thermoelasticity is extended to permit the calculation of stresses at both boundary and interior points. However, the new formulation still requires only surface discretization and is capable of very accurate evaluation of stress. The boundary displacement integral equation is successively differentiated to obtain the corresponding stress and strain integral equations. Explicit expressions for the steady-state axisymmetric fundamental solutions are derived in this process. The non-steady components of the integrands are obtained directly from the corresponding three-dimensional counterparts. The numerical implementation of this formulation is carried out within a general purpose BEM computer code and several numerical examples are presented to illustrate the accuracy and applicability of the method.

A boundary element method is developed for problems of quasistatic axisymmetric thermoelasticity. Unlike previous approaches, this new time domain formulation is written exclusively in terms of surface quantities, thereby eliminating the need for volume discretization. Furthermore, since the exact three-dimensional infinite space fundamental solutions are employed, very accurate solutions are obtainable, including the determination of surface stresses. In the integral formulation, the fundamental solutions are separated into steady-state and transient components. The steady-state portion, which contains all of the singularities, is integrated analytically in the circumferential direction, yielding the familiar axisymmetric kernels. The remaining non-singular transient integrands are treated by a combination of analytical and numerical quadrature. The method is implemented in a general purpose boundary element code, which includes multiregion capability along with higher order conforming surface elements. Several numerical examples are provided to illustrate the validity of the formulation and the attractiveness of this approach for practical engineenring analysis.

A boundary element formulation is presented for moderate Reynolds number, steady, incompressible, thermoviscous flows. The governing integral equations are written exclusively in terms of velocities and temperatures, thus eliminating the need for the computation of any gradients. Furthermore, with the introduction of reference velocities and temperatures, volume modeling can often be confined to only a small portion of the problem domain, typically near obstacles or walls. The numerical implementation includes higher order elements, adaptive integration and multiregion capability. Both the integral formulation and implementation are discussed in detail. Several examples illustrate the high level of accuracy that is obtainable with the current method.

An advanced boundary element method (BEM) is presented for the transient heat conduction analysis of engineering components. The numerical implementation necessarily includes higher-order conforming elements, self-adaptive integration and a multiregion capability. Planar, three-dimensional and axisymmetric analyses are all addressed with a consistent time-domain convolution approach, which completely eliminates the need for volume discretization for most practical analyses. The resulting general purpose algorithm establishes BEM as an attractive alternative to the more familiar finite difference and finite element methods for this class of problems. Several detailed numerical examples are included to emphasize the accuracy, stability and generality of the present BEM. Furthermore, a new efficient treatment is introduced for bodies with embedded holes. This development provides a powerful analytical tool for transient solutions of components, such as casting moulds and turbine blades, which are cumbersome to model when employing the conventional domain-based methods.

A general boundary element method (BEM) for the complete three-dimensional Biot consolidation theory is developed which operates directly in the time domain and requires only boundary discretization. Consequently, the dimensionality of the problem is reduced by one and the method becomes quite attractive for geotechnical analyses, particularly those that involve extensive or infinite domains. Furthermore, as a result of a well-known analogy fully elaborated here, the new BEM formulation is equally applicable for quasi-static thermoelasticity. Several detailed examples are presented to illustrate the accuracy and suitability of this boundary element approach for both consolidation and thermomechanical analyses.

The development of a time domain boundary element method for axisymmetric quasistatic poroelasticity is discussed. This new formulation, for the complete Biot consolidation theory, has the distinct advantage of being written exclusively in terms of boundary variables. Thus, no volume discretization is required, and the approach is ideally suited for geotechnicul problems involving media of infinite extent. In the presentation, the required axisymmetric integral equations and kernel functions are first developed from the corresponding three-dimensional theory. In particular, emphasis is placed on the analytical and numerical treatment of the kernels. This is followed by an overview of the numerical implementation, and a demonstration of its merits via the consideration of several examples.

Transient dynamic fracture mechanics problems are studied using an advanced two-dimensional formulation of the time-domain boundary element method. This formulation employs a set of well-behaved kernel functions, which have been used recently for the solution of general problems of transient elastodynamics. These kernels produce very stable results with respect to both time step and element size variations and consequently, are ideally suited for fracture mechanics studies. In the present implementation, both ordinary quarter-point and traction-singular quarter-point elements are developed. The dynamic stress intensity factors (DSIF) are then obtained from the crack opening displacements (COD) as well as from the crack-tip nodal tractions of the traction-singular elements. A number of example problems are investigated in detail. Included are convergence studies, comparisons with previously published results, and an evaluation of the various methods available for computing DSIF.

A boundary element method is developed for the solution of time-dependent problems in three-dimensional thermoelasticity. The time domain, boundary-only formulation represents the first of its kind for quasistatic analysis, which by definition considers transient heat conduction, but ignores the effects of inertia. By eliminating the need for volume discretization, the method becomes an attractive alternative to finite element analysis for this class of problems. Additionally, because an exact Green's function is used in the interior, steep thermal gradients can be captured much more readily than with standard domain-based methods. The presentation includes details of the fundamental solution, a derivation of the boundary integral equations, and an overview of a general purpose numerical implementation. This implementation permits the solution of large, multiregion problems with arbitrary geometry and boundary conditions. Several examples are included to validate the proposed method, as well as to highlight its usefulness.

We have recently developed a novel multi-level boundary element method (MLBEM) for steady heat diffusion in irregular two-dimensional domains (Numerical Heat Transfer Part B: Fundamentals, 46 : 329–356, 2004). This presentation extends the MLBEM methodology to three-dimensional problems. First, we outline a 3-D MLBEM formulation for steady heat diffusion and discuss the differences between multi-level algorithms for two and three dimensions. Then, we consider an example problem that involves heat conduction in a semi-infinite three-dimensional domain. We investigate the performance of the MLBEM formulation using a single-patch approach. The MLBEM algorithms are shown facilitate fast and accurate numerical solutions with no loss of the solution accuracy. More dramatic speed-ups can be achieved provided that patch-edge corrections are also evaluated using multi-level technique.Copyright © 2005 by ASME

Previous work on the development of multi-level boundary element methods (MLBEM) has produced dramatic computational efficiencies compared to traditional methods. However, for problems involving the oscillatory kernels associated with the Helmholtz equation, the performance improvements demonstrated to date have been limited due to the presence of coarse mesh Nyquist constraints. In the present paper, we examine an extension of the original Helmholtz MLBEM to remove these limitations, based on a polar decomposition of the kernels and field variables originally proposed by Brandt. In this initial numerical implementation of these ideas, the algorithm is applied to a pair of two-dimensional half-space problems and, therefore, is limited to the fast evaluation of Rayleigh integrals. In general, excellent performance is achieved. For example, in the second problem with a non-dimensional wave number equal to 105 , we obtain speed-up factors of approximately 300,000 versus a traditional BEM approach, while maintaining comparable levels of accuracy.Copyright © 2005 by ASME

Higher-order boundary element methods (BEM) are presented for three-dimensional steady convective heat diffusion at high Peclet numbers. An accurate and efficient boundary element formulation is facilitated by the definition of an influence domain due to convective kernels. This approach essentially localizes the surface integrations only within the domain of influence which becomes more narrowly focused as the Peclet number increases. The outcome of this phenomenon is an increased sparsity and improved conditioning of the global matrix. Therefore, iterative solvers for sparse matrices become a very efficient and robust tool for the corresponding boundary element matrices. In this paper, we consider an example problem with an exact solution and investigate the accuracy and efficiency of the higher-order BEM formulations for high Peclet numbers in the range from 1,000 to 100,000. The bi-quartic boundary elements included in this study are shown to provide very efficient and extremely accurate solutions, even on a single engineering workstation.Copyright © 2004 by ASME

In this presentation, we re-visit the poly-region boundary element methods (BEM) proposed earlier for the steady Navier-Stokes [1] and Boussinesq [2] flows, and develop a novel higher-order BEM formulation for the thermoviscous fluid flows that involves the definition of the domains of kernel influences due to steady Oseenlets. We introduce region-by-region implementation of the steady-state Oseenlets within the poly-region boundary element fequatramework, and perform integration only over the (parts of) higher-order boundary elements and volume cells that are influenced by the kernels. No integration outside the domains of the kernel influences are needed. Owing to the properties of the convective Oseenlets, the kernel influences are very local and propagate upstream. The localization becomes more prominent as the Reynolds number of the flow increases. This improves the conditioning of the global matrix, which in turn, facilitates an efficient use of the iterative solvers for the sparse matrices [3]. Here, we consider quartic boundary elements and bi-quartic volume cells to ensure a high level resolution in space. Similar to the previous developments [4–6], coefficients of the discrete boundary integral equations are evaluated with the sufficient precision using semi-analytic approach to ensure exceptional accuracy of the boundary element formulation. To demonstrate the attractiveness of the poly-region BEM formulation, we consider a numerical example of the well-known Rayleigh-Benard problem governed by the Boussinesq equations.Copyright © 2004 by ASME

A new space-time multi-level boundary element method (MLBEM) is developed for transient heat diffusion problems in two-dimensions. This approach extends the MLBEM approach for steady heat diffusion [1] to accommodate fast time convolution algorithm [2]. The space-time MLBEM algorithm developed in this presentation provides fast, accurate and memory efficient numerical solutions for time-dependent heat diffusion problems. Conventional BEM approaches using M boundary elements result in operation counts of order O(M2N2) 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(M log MN3/2) for a two-dimensional model problem using the multi-level boundary element algorithm. Memory requirements are also significantly reduced, while maintaining the same level of accuracy as the conventional time domain BEM approach.

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(N3/2 ) for a couple of 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 © 2004 by ASME

Recently, we have developed multi-level boundary element methods (MLBEM) for the solution of the Laplace and Helmholtz equations that involve asymptotically decaying non-oscillatory and oscillatory singular kernels, respectively. The accuracy and efficiency of the fast boundary element methods for steady-state heat diffusion and accoustics problems have been investigated for square domains. The current work extends the MLBEM methodology to the solution of Stokes equation in more complex two dimensional domains. The performance of the fast boundary element method for the Stokes flows is first investigated for a model problem in a unit square. Then, we study the performance of the MLBEM algorithm in a C-shaped domain.Copyright © 2004 by ASME