Shape design sensitivity analysis of kinematical boundaries

A new approach is used in this paper to derive the design sensitivity formulation with kinematical design boundaries. By employing the concept of the conventional finite difference approach, the variation of structural response due to change of the kinematic design boundary can be represented by the perturbed structure under a set of kinematical boundary conditions. Parameterization of the design variation with respect to the design variable enables us to transform the design sensitivity into the solutions of a boundary value problem with perturbation displacements on the design boundary. The perturbation diplacements can be evaluated from the stress and displacement fields of the initial problem. This approach can be treated as a special case of the general direct formulation, but the derivation using the finite difference procedure gives a strong physical meaning of the method, and the formulation derived provides an explicit form for design sensitivity calculation. The numerical implementation of this approach based on the boundary element method is discussed, and a few numerical examples are used to verify the proposed formulation.     

A new BEM approach for reissner's plate bending

A new boundary element formulation for Reissner's plate bending is presented. This form of BEM has an advantage in that the bending stresses on the boundary can be calculated directly from the numerical solution, avoiding the use of tangential derivatives of displacement for finding plate bending stresses on the boundary. The effectiveness of the approach is also discussed through some test examples. In the present BEM formulation, the singular orders of the two kernels are the same as those in the standard BEM formulation of a Reissner's type plate—one of which is logarithmic singular and the other is 1/r singular.     

Variational formulation of linear problems with nonhomogeneous boundary conditions and internal discontinuities

A variational formulation applicable to linear operators with nonhomogeneous boundary conditions and jump discontinuities is presented. For the formulation to be applicable, the boundary condition and discontinuities have to be consistent with the operator governing the field problem. The problem is set up in a space of suitable continuous bilinear mapping. Thus, operators on inner product spaces, convolution spaces and energy spaces are included as specializations. The basic construction can be used to generate dual-complementary variational principles. Implementation is illustrated by examples. The role of boundary terms in finite element discretizations based on interpolants of limited smoothness is discussed.     

Application of a bounding surface model to Boston blue clay

A formulation for an elastic-plastic Bounding Surface model for soils is presented. The model is calibrated and the predicted and measured response compared for laboratory stress paths and a boundary value problem. The predictions are reasonable, providing confidence in the model formulation, numeric implementation and numeric solution techniques. Possible improvements to the model are discussed.     

Vibration analysis of continuous plate structures using boundary integrals

The natural frequencies of thin continuous plates, including the influence of in-plane forces, are calculated using boundary integrals. This study deals in particular with continuous plate structures consisting of several fields of different thickness and possibly with the inclusion of internal supports. The concepts of compatibility and equilibrium conditions along the common boundaries have been imposed in conjunction with a boundary element formulation for each panel. Several examples are presented and the results are compared with analytical as well as finite element solutions demonstrating the effectiveness of the formulation. Some numerical issues are discussed as well.     

A unified approach to the mathematical analysis of generalized RKPM,gradient RKPM,and GMLS

It is well-known that the conventional reproducing kernel particle method (RKPM) is unfavorable when dealing with the derivative type essential boundary conditions [1], [2], [3]. To remedy this issue a group of meshless methods in which the derivatives of a function can be incorporated in the formulation of the corresponding interpolation operator will be discussed. Formulation of generalized moving least squares (GMLS) on a domain and GMLS on a finite set of points will be presented. The generalized RKPM will be introduced as the discretized form of GMLS on a domain. Another method that helps to deal with derivative type essential boundary conditions is the gradient RKPM which incorporates the first gradients of the function in the reproducing equation. In present work the formulation of gradient RKPM will be derived in a more general framework. Some important properties of the shape functions for the group of methods under consideration are discussed. Moreover error estimates for the corresponding interpolants are derived. By generalizing the concept of corrected collocation method, it will be seen that in the case of employing each of the proposed methods to a BVP, not only the essential boundary conditions involving the function, but also the essential boundary conditions which involve the derivatives could be satisfied exactly at particles which are located on the boundary.     

Wideband and Isotropic Room Acoustics Simulation Using 2-D Interpolated FDTD Schemes

In this paper, a complete method for finite-difference time-domain modeling of rooms in 2-D using compact explicit schemes is presented. A family of interpolated schemes using a rectilinear, nonstaggered grid is reviewed, and the most accurate and isotropic schemes are identified. Frequency-dependent boundaries are modeled using a digital impedance filter formulation that is consistent with locally reacting surface theory. A structurally stable and efficient boundary formulation is constructed by carefully combining the boundary condition with the interpolated scheme. An analytic prediction formula for the effective numerical reflectance is given, and a stability proof provided. The results indicate that the identified accurate and isotropic schemes are also very accurate in terms of numerical boundary reflectance, and outperform directly related methods such as Yee's scheme and the standard digital waveguide mesh. In addition, one particular scheme-referred to here as the interpolated wideband scheme-is suggested as the best scheme for most applications.     

Reconstruction of heat transfer coefficients using the boundary element method

This paper describes the reconstruction of the heat transfer coefficient (space, Problem I, or time dependent, Problem II) in one-dimensional transient inverse heat conduction problems from surface temperature or average temperature measurements. Since the inverse problem posed does not involve internal temperature measurements, this means that non-destructive testing of materials can be performed. In the formulation, convective boundary conditions relate the boundary temperature to the heat flux. Numerical results obtained using the boundary element method are presented and discussed.     

An important note on symmetry line boundary conditions in fibre-reinforced laminated anisotropic composites

Symmetry line boundary conditions of symmetrical and unsymmetrical laminated fibrereinforced anisotropic composite structures are discussed. A simple C⁰ isoparametric finite element formulation based on the Reissner-Mindlin first-order shear deformation theory is used. The effects of boundary conditions on the laminated scheme and the orientation of fibres are studied. Numerical results for deflections presented herein should be of interest to composite-structure designers, experimentalists, and numerical analysts in verifying their results.     

Trefftz direct method

This paper discusses a so-called Trefftz Direct Method (TDM), in which the non-singular boundary integral equation is used as the starting formulation and complete sets of solution are used as weighting functions. Several relevant characteristics and problems of this approximation are also discussed.     

A unified treatment of boundary conditions in least-square based finite-volume methods

We propose a unified treatment of internal and boundary vertex least-squares reconstructions in second-order accurate cell-centered finite-volume discretisation of 2-D steady diffusion problems. Dirichlet, Neumann, and Robin boundary conditions are taken into account in the same formulation by introducing suitable constraints in the least-squares minimization process. The method is discussed in its theoretical framework and a representative numerical experiment illustrates its capability in providing the second-order of accuracy.     

Viscous/inviscid interaction procedures for compressible aerodynamic flow simulations

Steady transonic flows over a wing are calculated based on a hierarchical formulation where the potential flows are corrected due to the entropy and vorticity effects of shock waves and/or laminar boundary layers. Preliminary results are in agreement with standard Euler and Navier-Stokes calculations. The merits of the present approach are briefly discussed.     

Size-dependent optimal microstructure design based on couple-stress theory

The purpose of this paper is to propose a size-dependent topology optimization formulation of periodic cellular material microstructures, based on the effective couple-stress continuum model. The present formulation consists of finding the optimal layout of material that minimizes the mean compliance of the macrostructure subject to the constraint of permitted material volume fraction. We determine the effective macroscopic couple-stress constitutive constants by analyzing a unit cell with specified boundary conditions with the representative volume element (RVE) method, based on equivalence of strain energy. The computational model is established by the finite element (FE) method, and the design density and FE stiffness of the RVE are related by the solid isotropic material with penalization power (SIMP) law. The required sensitivity formulation for gradient-based optimization algorithm is also derived. Numerical examples demonstrate that this present formulation can express the size effect during the optimization procedure and provide precise topologies without increase in computational cost.     

Dynamic inelastic structural analysis by boundary element methods

Summary Boundary element methodologies for the determination of the response of inelastic two-and three-dimensional solids and structures as well as beams and flexural plates to dynamic loads are briefly presented and critically discussed. Elastoplastic and viscoplastic material behaviour in the framework of small deformation theories are considered. These methodologies can be separated into four main categories: those which employ the elastodynamic fundamental solution in their formulation, those which employ the elastostatic fundamental solution in their formulation, those which combine boundary and finite elements for the creation of an efficient hybrid scheme and those representing special boundary element techniques. The first category, in addition to the boundary discretization, requires a discretization of those parts of the interior domain expected to become inelastic, while the second category a discretization of the whole interior domain, unless the inertial domain integrals are transformed by the dual reciprocity technique into boundary ones, in which case only the inelastic parts of the domain have to be discretized. The third category employs finite elements for one part of the structure and boundary elements for its remaining part in an effort to combine the advantages of both methods. Finally, the fourth category includes special boundary element techniques for inelastic beams and plates and symmetric boundary element formulations. The discretized equations of motion in all the above methodologies are solved by efficient step-by-step time integration algorithms. Numerical examples involving two-and three-dimensional solids and structures and flexural plates are presented to illustrate all these methodologies and demonstrate their advantages. Finally, directions for future research in the area are suggested.     

On a Galerkin boundary node method for potential problems

A Galerkin boundary node method (GBNM), for boundary only analysis of partial differential equations, is discussed in this paper. The GBNM combines an equivalent variational form of a boundary integral equation with the moving least-squares (MLS) approximations for generating the trial and test functions of the variational formulation. In this approach, only a nodal data structure on the boundary of a domain is required, and boundary conditions can be implemented directly and easily despite of the fact that the MLS shape functions lack the delta function property. Formulations of the GBNM using boundary singular integral equations of the second kind for potential problems are developed. The theoretical analysis and numerical results indicate that it is an efficient and accurate numerical method.     

Boundary conditions in mixture theory and in CFD applications of higher order models

We discuss the importance of and the need for (additional) boundary conditions in Mixture Theory (also known as the Theory of Interacting Continua). Specifically, we will give an overview of the model due to Rajagopal and Massoudi which is appropriate for the flow of a linearly viscous fluid infused with solid particles. The solid particles are modeled as granular materials. In this formulation the need for additional boundary condition arises due to higher gradients of density (or volume fraction). The challenging issue of how to ‘split’ the total stress or the total velocity at the boundary is also discussed.     

Boundary element solution for elastic orthotropic half-plane problems

This paper presents the boundary element solution for orthotropic half-plane problems. The complete fundamental solutions due to unit point loads within the half-plane are given. The boundary integral formulation using these fundamental solutions is presented. Expressions for stresses and displacements at internal points are also given. This formulation is applied to some classical problems. This solution procedure is highly accurate and computationally more efficient than the boundary element formulation using the Kelvin fundamental solution for orthotropic half-plane problems.     

Assessment of potential-based fluid finite elements for seismic analysis of dam–reservoir systems

This paper assesses the use of a potential-based fluid finite element formulation to investigate earthquake excited dam–reservoir systems. The mathematical background of the analytical and numerical techniques is presented in a unified format. Frequency and time-domain analyses are conducted to validate the potential-based finite element formulation. A case study of a typical dam–reservoir system subjected to earthquake loading is presented. The dynamic response of the system is discussed to illustrate the effects of fluid–structure interaction and reservoir bottom absorption. The validated potential-based fluid elements and boundary conditions are shown to perform adequately for practical seismic analysis of dam–reservoir systems.     

Numerical approximation of incompressible flows with net flux defective boundary conditions by means of penalty techniques

We consider incompressible flow problems with defective boundary conditions prescribing only the net flux on some inflow and outflow sections of the boundary. As a paradigm for such problems, we simply refer to Stokes flow. After a brief review of the problem and of its well posedness, we discretize the corresponding variational formulation by means of finite elements and looking at the boundary conditions as constraints, we exploit a penalty method to account for them. We perform the analysis of the method in terms of consistency, boundedness and stability of the discrete bilinear form and we show that the application of the penalty method does not affect the optimal convergence properties of the finite element discretization. Since the additional terms introduced to account for the defective boundary conditions are non-local, we also analyze the spectral properties of the equivalent algebraic formulation and we exploit the analysis to set up an efficient solution strategy. In contrast to alternative discretization methods based on Lagrange multipliers accounting for the constraints on the boundary, the present scheme is particularly effective because it only mildly affects the structure and the computational cost of the numerical approximation. Indeed, it does not require neither multipliers nor sub-iterations or additional adjoint problems with respect to the reference problem at hand.