Shape optimization of two-dimensional thermal conducting solid using boundary integral equation formulation

A novel boundary integral formulation is presented for the direct solution of the classical problem of slow flow past a two-dimensional cylinder of arbitrary cross section in an unbounded viscous medium, the equations of motion having first been linearised by the Oseen approximation. It is shown how the governing partial differential equations of motion, together with the no-slip boundary conditions on the cylinder, may be reformulated as a pair of coupled integral equations of the second kind, which may be manipulated further to yield the lift and drag coefficients explicitly, as well as flow characteristics anywhere in the flowfield. The present formulation requires a non-iterative numerical solution procedure which is applicable to low Reynolds number flows. The method is not restricted in its ability to deal with complicated cylinder geometries, as the discretisation of only the cylinder surface is required. Results of the present method are shown to be in good agreement with those of previous analytical and numerical investigations.

The objective of this study is to examine two analytical methods for the design sensitivity analysis of optimal control problems and to develop the analyses necessary for the application of the methods to thermal systems. The two methods are the direct differentiation method and the adjoint variable method. Results for these methods are compared with those obtained using central finite difference approximations for design sensitivity analysis. The adjoint variable method has the potential to reduce significantly computational times for design sensitivity analysis while sacrificing only a small amount of accuracy. As the number of functional requiring design sensitivity analysis increases, however, the adjoint variable method becomes less attractive and the direct differentiation method becomes more attractive.

An adjoint approach for the shape design sensitivity analysis (SDSA) of plane elasticity problems with a direct boundary integral equation formulation is presented. The objective function is expressed in an integral form and augmented by incorporating the elasticity equations via adjoint functions. The variation of the augmented function is taken by performing its material derivative to obtain the sensitivity of the objective function. The adjoint functions are obtained through the solution of the adjoint problem resulting from setting the local variations of the displacement, velocity, and traction fields equal to zero. An approximate procedure for the application of concentrated adjoint load in the boundary element framework is suggested. This allows the computation of displacement, traction, and stress sensitivities at discrete nodal points. The present formulation includes the treatment of body forces in sensitivity calculations. Numerical examples demonstrate the validity of the present approach.

Based on the Rizzo’s direct boundary integral equation formulation for elasticity problems, elastic bodies with randomly distributed circular inclusions are simulated using the boundary element method. The given numerical examples show that the boundary element method is more accurate and more efficient than the finite element method for such type of problems. The presented approach can be successfully applied to estimate the equivalent elastic properties of many composite materials.

This paper deals with shape optimization of three-dimensional elastic bodies. The proposed approach is based on natural design variables and shape functions and utilizes the boundary element method. The design variables are the magnitudes of a set of fictitious loads applied on the structure. Shape optimal design problems based on minimum mean compliance, peak stress minimization and stress constraints are considered. A general method for shape design sensitivity analysis using the material derivative concept and adjoint variable method is used. The sensitivity formula for a general stress constraint imposed over the optimized boundary is derived. Optimal shapes for three-dimensional problems are presented to show numerical applications.

An integral equation formulation of anti-plane inhomogeneities

Design sensitivity analysis for transient response of non-viscously damped systems based on direct differentiate method

On the elastic double layer: Some exact solutions and the spectrum on the sphere

Using the bond-based peridynamics and the parallel computation with binary decomposition, an adjoint shape design sensitivity analysis(DSA) method is developed for the dynamic crack propagation problems. The peridynamics includes the successive branching of cracks and employs the explicit scheme of time integration. The adjoint variable method is generally not suitable for path-dependent problems but employed since the path of response analysis is readily available. The accuracy of analytical design sensitivity is verified by comparing it with the finite difference one. The finite difference method is susceptible to the amount of design perturbations and could result in inaccurate design sensitivity for highly nonlinear peridynamics problems with respect to the design. It turns out that C1-continuous volume fraction is necessary for the accurate evaluation of shape design sensitivity in peridynamic discretization.

This paper is concerned with the development of an exterior domain segmentation method for the solution of two‐ or three‐dimensional time‐harmonic scattering problems in acoustic media. This method, based on a variational localized, symmetric, boundary integral equation formulation leads, upon discretization, to a sparse system of algebraic equations whose solution requires only O(N) multiplications, where N is the number of unknown nodal pressures on the scatterer surface. The new procedure is analogous to the one developed recently1 except that in the present formulation we avoid completely the use of the hypersingular operator, thereby reducing the computational complexity. Numerical experiments for a rigid circular cylindrical scatterer subjected to a plane incident wave serve to assess its accuracy for normalized wave numbers, ka, ranging from 0 to 30, both directly on the scatterer and in the far field, and to confirm that, contrary to standard boundary integral equation formulations, the present procedure is valid for critical frequencies.

New adjoint variable method for design sensitivity analysis of distinct eigenvlaues and eigenvectors is presented. In the viewpoint of efficiency for the design sensitivity analysis of eigenvectors especially, the developed adjoint variable method is required to compute adjoint variables from simultaneous linear equations, the so-called adjoint equations, instead of linear combination of eigenvectors. Once we obtain the adjoint variables, design sensitivity analysis of response function that is given in terms of eigenvalues, eigenvectors and design variables can be computed directly. In this way, design sensitivity analysis of eigenvectors can be obtained by using eigenvalues and their corresponding eigenvectors of the mode being differentiated only. To verify the proposed method, numerical examples are demonstrated. This can have considerable impact on computer implementation of the developed method in the design sensitivity analysis of eigenproblem needed for practical applications.

The Earth's magnetic field is sustained by dynamo action in the fluid outer core. The energy sources available to the geodynamo are well established, but their relative importance remains uncertain. We focus on the issue of thermal versus compositional convection, which is inextricably coupled to the evolution of the core as the Earth cools. To investigate the effect of the various physical processes on this evolution, we develop models based on conservation of energy and the assumption that the core is well mixed by vigorous convection. We depart from previous numerical studies by developing an analytical model. The simple algebraic form of the solution affords insight into both the evolution of the core and the energy budget of the geodynamo. We also present a numerical model to compare with the quantitative predictions of our analytical model and find that the differences between the two are negligible. An important conclusion of this study is that thermal convection can contribute significantly to the geodynamo. In fact, a modest heat flux in excess of that conducted down the adiabatic gradient is sufficient to power the geodynamo, even in the absence of compositional convection and latent‐heat release. The relative contributions of thermal and compositional convection to the dynamo are largely determined by the magnitude of the heat flux from the core and the inner‐core radius. For a plausible current‐day heat flux of Q = 3.0 × 1012 W and the current inner‐core radius, we find that compositional convection is responsible for approximately two thirds of the ohmic dissipation in the core and thermal convection for the remaining one third. The proportion of ohmic dissipation produced by thermal convection increases to 45% with an increase in Q to 6.0 × 1012 W. In the early Earth, when the inner core was smaller and the heat flux probably greater than the present values, thermal convection would have been the dominant energy source for the dynamo. We also calculate the history of inner‐core growth as a function of the heat flux. For example, the inner core would have grown to its present size in 2.8 × 109 years if the average heat flux was Q = 4.0 × 1012 W. The model does not require the heat flux to be constant.

A face based basis function solutions to thermal conduction problems for 3-D irregular shaped bodies using BEM

Details are given of the development and application of a 2D depth-integrated, conformal boundary-fitted, curvilinear model for predicting the depth-mean velocity field and the spatial concentration distribution in estuarine and coastal waters. A numerical method for conformal mesh generation, based on a boundary integral equation formulation, has been developed. By this method a general polygonal region with curved edges can be mapped onto a regular polygonal region with the same number of horizontal and vertical straight edges and a multiply connected region can be mapped onto a regular region with the same connectivity. A stretching transformation on the conformally generated mesh has also been used to provide greater detail where it is needed close to the coast, with larger mesh sizes further offshore, thereby minimizing the computing effort whilst maximizing accuracy. The curvilinear hydrodynamic and solute model has been developed based on a robust rectilinear model. The hydrodynamic equations are approximated using the ADI finite difference scheme with a staggered grid and the solute transport equation is approximated using a modified QUICK scheme. Three numerical examples have been chosen to test the curvilinear model, with an emphasis placed on complex practical applications.

We present a comparison among the performance of solvers based on Nystr\om discretizations of several well-posed boundary integral equation formulations of Helmholtz transmission problems in two-dimensional Lipschitz domains. Specifically, we focus on the following four classes of boundary integral formulations of Helmholtz transmission problems: (1)~the classical first kind integral equations for transmission problems~\cite {costabel-stephan}, (2)~the classical second kind integral equations for transmission problems~\cite {KressRoach}, (3)~the single integral equation formulations~\cite {KleinmanMartin}, and (4)~certain direct counterparts of recently introduced generalized combined source integral equations \cite {turc2, turc3}. The former two formulations were the only formulations whose well-posedness in Lipschitz domains was rigorously established \cite {costabel-stephan, ToWe:1993}. We establish the well-posedness of the latter two formulations in appropriate functional spaces of boundary traces of solutions of transmission Helmholtz problems in Lipschitz domains. We give ample numerical evidence that Nystr\om solvers based on formulations (3) and (4) are computationally more advantageous than solvers based on the classical formulations (1) and (2), especially in the case of high-contrast transmission problems at high frequencies.