Shape design optimization of thermoelasticity problems using isogeometric boundary element method

We derive the isogeometric configuration sensitivity of the Mindlin plates by using the material derivative and adjoint approaches. This is utilized in the shape design optimization that includes a variation of design components in its shape and orientation. By the isogeometric approach, the NURBS basis function in CAD system is directly utilized in the response analysis, which enables the seamless incorporation of higher continuity and exact geometry such as curvature and normal vector into the computational framework. The impact of exact curvature in the bending problem of Mindlin plates on the configuration design sensitivity is demonstrated through numerical examples. The obtained design sensitivity is further utilized in the shape design optimization of built-up structures. Due to the non-interpolatory property of the NURBS basis functions, a mismatch of patches in the built-up structures could occur during the isogeometric design optimization, which can be easily resolved using transformed basis functions.

Direct differentiation formulation for boundary element shape sensitivity analysis of axisymmetric elastic solids

The one- and two-dimensional inviscid Euler equations are formulated in an integro-differential form for shape design sensitivity analysis and optimization. The principal tool employed to derive the performance derivative sensitivity equations is the variational method, which is a continuous alternative to the discrete sensitivity analysis. Along with the sensitivity equations, the co-state equations and their boundary conditions are derived. Thereafter, based on the modal analysis of Von Neumann theory, the stability limits of the co-state equations are investigated. The stability characteristics of the co-state equations are compared with those of the Euler equations. Finally, using the criteria from the stability analysis of the state and co-state equations, some results of shape design optimization and sensitivity analysis for both quasi one- and two-dimensional Euler equations are presented. Copyright © 1999 John Wiley & Sons, Ltd.

Non-equivalence of the conventional boundary integral formulation and its elimination for plane elasticity problems

An isogeometric boundary element method is applied to simulate wave scattering problems governed by the Helmholtz equation. The NURBS (non-uniform rational B-splines) widely used in the CAD (computer aided design) field is applied to represent the geometric model and approximate physical field variables. The Burton-Miller formulation is used to overcome the fictitious frequency problem when using a single Helmholtz boundary integral equation for exterior boundary-value problems. The singular integrals existing in Burton-Miller formulation are evaluated directly and accurately using Hadamard’s finite part integration. Fast multipole method is applied to accelerate the solution of the system of equations. It is demonstrated that the isogeometric boundary element method based on NURBS performs better than the conventional approach based on Lagrange basis functions in terms of accuracy, and the use of the fast multipole method both retains the accuracy for isogeometric boundary element method and reduces the computational cost.

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.

An isogeometric approach of two dimensional acoustic design sensitivity analysis and topology optimization analysis for absorbing material distribution

Non-stationary boundary value problems of uncoupled thermoelasticity are considered.             A method of boundary integral equations (BIME) in the initial space-time has been developed for solving non-stationary boundary value problems of thermoelasticity under plane deformation. On the basis of the method of generalized functions, generalized solutions of boundary value problems are constructed using the Green's function for the heat equation and the Green's tensor of the Lame equations under the action of non-stationary power and heat sources of various types. Integral representations of the solution of boundary value problems are obtained. These solutions allow, based on known boundary values and initial conditions (displacements, temperature, stresses and heat flux), to determine the thermally stressed state of the medium under the influence of various power and thermal loads. Resolving boundary integral equations are constructed to determine the unknown boundary functions.

On the conventional boundary integral equation formulation for piezoelectric solids with defects or of thin shapes

We consider the conductivity transmission problem in two dimensions with a simply connected inclusion of arbitrary shape. It is well known that the solvability of the transmission problem can be established via the boundary integral formulation in which the Neumann--Poincaré (NP) operator is involved. In this paper, we derive series expansions of the layer potential operators based on geometric function theory and exhibit a novel approach to the transmission problem. We first construct a collection of harmonic basis functions by using the Faber polynomials associated with the simply connected inclusion. We then derive explicit series expansions of the layer potential operators with respect to the constructed basis functions. In particular, the NP operator becomes a doubly infinite, self-adjoint matrix operator, whose entry is explicitly given by the Grunsky coefficients corresponding to the inclusion. By applying the finite section method to this matrix formulation, we obtain an approximation scheme for the spectrum of the NP operator and the solution to the transmission problem. This scheme allows us to numerically compute, in a simple way, the spectrum of the NP operator for a smooth domain when the exterior conformal mapping is known. We additionally derive an explicit integral expression for the coefficients of the exterior conformal mapping so that for an inclusion of arbitrary shape, one can numerically compute the exterior conformal mapping by solving only one boundary integral equation and apply the proposed approach. We provide numerical examples to demonstrate the effectiveness of the proposed method.

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

Frequency-robust preconditioning of boundary integral equations for acoustic transmission

A method for shape design sensitivity analysis of two-dimensional thermal conducting solid is presented using the material derivative concept and the adjoint variable method. A general thermal boundary condition with heat convection is considered in addition to prescribed temperature and heat flux. The method for deriving the sensitivity formula is based on standard direct boundary integral equation formulation. The sensitivity of a general functional depending on temperature and heat flux is considered. The method is then applied to obtain the sensitivity analysis is demonstrated by a hollow cylinder problem with exact solution. A weight minimization problem of a thermal diffuser is considered as a practical application. The sensitivity by the presented method is compared with that by finite differences and an optimal shape is found by use of an optimization routine.

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 problem of determination of the stationary temperature field in a body with plane sheet of heat sources is reduced to the solution of an integral equation of the first kind. A method for the determination of the set of solutions to this equation is proposed. The components of the vector of elastic displacements and the components of the tensor of temperature stresses are found according to the known temperature field by using the equations of thermoelasticity. We study the distributions of thermal displacements and stresses for any axially symmetric distribution of temperature in a circular domain. Numerous elements of contemporary engineering constructions operate under conditions of nonuniform heating that promote the formation of temperature gradients and thermal stresses. For the comprehensive analysis of the strength of structures, it is necessary to know the levels of thermal stresses and the character of their action. The investigation of the thermoelastic state of heat-generating elements is of high importance for practical purposes [9]. The presence of thin heat-generating inclusions in a body leads to the formation of tensile or compressive stresses. In the first case, the action of these stresses, together with the action of the other factors, may lead to crack initiation and propagation. In the second case, the action of stresses may decrease the intensity of stresses in the vicinity of the existing crack under force loading and promote the formation of conditions inhibiting the process of crack growth [10]. In the investigation of the stressed state of a body with thermally active cracks (with temperature or heat fluxes given on these cracks), the determination of stresses acting at the sites of the cracks is an intermediate stage and then the stresses acting upon the crack surfaces are removed. The major part of investigations of the stress-strain state of bodies with circular thermally active or heatinsulated cracks are performed for axially symmetric problems by the method of integral Hankel transformation and dual integral equations [11–16]. In [3], a method of two-dimensional boundary integral equations based on the theory of the Newton potential is developed for the solution of three-dimensional problems of stationary heat conduction and thermoelasticity for bodies with cracks. According to this method, the densities of potentials in the problem of heat conduction are the intensities of heat sources at the sites of the cracks determined, for given temperatures on the cracks, from the integral equations with polar kernels. The temperature field and stressed state of a body with given temperature or heat flow in a circular domain described by polynomials of the third degree are investigated in [4]. In the axially symmetric case, for polynomials of any degree, this problem is studied in [5]. According to the classic statement of the problems of stationary heat conduction with thermally active inclusions [4, 5], the presence of heat sources is postulated solely in the domain of the inclusion, which leads to singular distributions of the heat flows on its edges.