A comparison of the continuous and discrete adjoint approach extended based on the standard lattice Boltzmann method in flow field inverse optimization problems

The objective of this research is the numerical implementation and comparison between the performance of the continuous and discrete adjoint Lattice Boltzmann (LB) methods in optimization problems of unsteady flow fields. For this purpose, a periodic two-dimensional incompressible channel flow affected by the constant and uniform body forces is considered as the base flow field. The standard LB method and D2Q9 model are employed to solve the flow field. Moreover, the inverse optimization of the selected flow field is defined by considering the body forces as the design variables and the sum of squared errors of flow field variables on the whole field as the cost function. In this regard, the continuous and discrete adjoint approaches extended based on the LB method are used to achieve the gradients of the cost function with respect to the design variables. Finally, the numerical results obtained from the continuous adjoint LB method are compared with the discrete one, and the accuracy and efficiency of them are discussed. In addition, the validity of the obtained cost function gradients is investigated by comparing with the results of the standard forward finite difference and complex step methods. The numerical results show that regardless of the implementation cost of the two approaches, the computational cost to evaluate the gradients in each optimization cycle for the discrete adjoint LB approach is slightly more than the other one but has a little higher convergence rate and needs a smaller number of cycles to converge. Besides, the gradients obtained from the discrete version have a better agreement with those of the complex step method. Eventually, based on the structural similarities of the continuous LB equation and its corresponding adjoint one and using the simple periodic and complete bounce-back boundary conditions for the LB equation, the improved boundary conditions for the continuous adjoint LB equation are presented. The numerical results show that the use of these boundary conditions instead of the original adjoint boundary conditions significantly improves the relative accuracy and also the convergence rate of the continuous adjoint LB method.     

Solution to the topology optimization problem using a time-evolution equation

This article describes a numerical solution to the topology optimization problem using a time-evolution equation. The design variables of the topology optimization problem are defined as a mathematical scalar function in a given design domain. The scalar function is projected to the normalized density function. The adjoint variable method is used to determine the gradient defined as the ratio of the variation of the objective function or constraint function to the variation of the design variable. The variation of design variables is obtained using the solution of the time-evolution equation in which the source term and Neumann boundary condition are given as a negative gradient. The distribution of design variables yielding an optimal solution is obtained by time integration of the solution of the time-evolution equation. By solving the topology optimization problem using the proposed method, it is shown that the objective function decreases when the constraints are satisfied. Furthermore, we apply the proposed method to the thermal resistance minimization problem under the total volume constraint and the mean compliance minimization problem under the total volume constraint.     

基于虚载荷变量的无网格法形状优化的研究

将选择施加在"虚结构"控制点上的虚载荷作为形状优化的设计变量,并将它与无网格Galerkin法相结合来开展结构形状优化研究,采用罚函数法来施加边界条件,通过直接微分法建立了结构形状优化的离散型灵敏度分析算法,利用无网格法研究了节点坐标关于设计变量导数的计算。所提出的算法简单明了,它不仅解决了网格的畸变问题,而且简化了优化模型和迭代流程,并可使结构的受力特性得到进一步的改善。最后用2个工程实例验证了所建立的算法,并得到了形状优化结果。     

Shape optimization using time evolution equations

In this study, we deal with a numerical solution based on time evolution equations to solve the optimization problem for finding the shape that minimizes the objective function under given constraints. The design variables of the shape optimization problem are defined on a given original domain on which the boundary value problems of partial differential equations are defined. The variations of the domain are obtained by the time integration of the solution to derive the time evolution equations defined in the original domain. The shape gradient with respect to the domain variations are given as the Neumann boundary condition defined on the original domain boundary. When the constraints are satisfied, the decreasing property of the objective function is guaranteed by the proposed method. Furthermore, the proposed method is used to minimize the heat resistance under a total volume constraint and to solve the minimization problem of mean compliance under a total volume constraint.     

考虑瞬态响应的薄板结构阻尼材料层拓扑优化

讨论了敷设阻尼材料的薄板结构考虑瞬态响应时阻尼材料层的最优布局问题。基于SIMP方法构造人工阻尼材料惩罚模型和结构拓扑优化模型,以阻尼材料的相对密度作为设计变量,在给定阻尼材料用量的条件下,最小化结构瞬态位移响应的时间积分。由于结构整体呈现非比例阻尼特性,采用逐步积分法对结构的振动方程进行求解。通过伴随变量法得到目标函数对设计变量的灵敏度表达式,在此基础上采用基于梯度的移动渐近线方法求解。数值算例验证了优化模型与算法的合理性和有效性。     

Design sensitivity analysis and optimization of interface shape for zoned-inhomogeneous thermal conduction problems using boundary integral formulation

A generalized formulation of the shape design sensitivity analysis for two-dimensional steady-state thermal conduction problem as applied to zoned-inhomogeneous solids is presented using the boundary integral and the adjoint variable method. Shape variation of the external and zone-interface boundary is considered. Through an analytical example, it is proved that the derived sensitivity formula coincides with the analytic solution. In numerical implementation, the primal and adjoint problems are solved by the boundary element method. Shape sensitivity is numerically analyzed for a compound cylinder, a thermal diffuser and a cooling fin problem, and its accuracy is compared with that by numerical differentiation. The sensitivity formula derived is incorporated to a nonlinear programming algorithm and optimum shapes are found for the thermal diffuser and the cooling fin problem.     

On the optimum shape of fillets in plates subjected to multiple in-plane loading cases

The shape of a plate in plane stress is determined, such that the maximum elastic stress corresponding to given loads is minimized. The shape of the boundary is approximated by a series and optimization is carried out by solving a sequence of linearized minimum–maximum problems using linear programming. The optimization problem is extended to include multiple loading cases and geometrical constraints. The stress derivatives are found using analytical expressions for stiffnesses and stiffness derivatives of the finite elements. For the optimum design of a hole in an infinite plate under biaxial stretching the numerical result is compared with an analytical solution. As another example the method is used to optimize the edge shape of a shape of a junction in a web frame.     

Shape design sensitivity analysis for non-linear anisotropic heat conducting solids and shape optimization by the BEM

Shape design sensitivity analysis (SDSA) expressions have been derived for non-linear anisotropic heat conducting solid bodies by following the material derivative concept and adjoint variable method of optimal shape design given in the literature. The variation of a general integral functional has been described in terms of primary and adjoint quantities evaluated at the varying boundaries. As an example problem in shape optimization, optimal outer boundary profiles of an orthotropic solid body are obtained by the boundary element method (BEM), after reformulating the SDSA equations in a form which is most suitable for the BEM.     

Topology optimization for thermo-mechanical compliant actuators using mesh-free methods

This article presents an alternative topology optimization method for the design of compliant actuators using mesh-free methods, in which the thermo-mechanical multi-physics modelling and geometrically non-linear analysis are included. The relatively new mesh-free method rather than the standard finite element method (FEM) is used to discretize the design domain and interpolate the bulk density field, because the mesh-free method is in some cases more capable of modelling the large-displacement compliant mechanisms involving the geometrical non-linearity. An interpolation scheme is used to indicate the dependence of material properties on element pseudo densities which are distributed to the corresponding integration points, and the method for imposing essential boundary conditions in mesh-free methods is also discussed. Furthermore, the adjoint approach is incorporated into the mesh-free method to perform the design sensitivity analysis. The optimization problem is established mathematically as a non-linear programming problem to which a sequential convex programming method is applied. The effectiveness of the proposed method is demonstrated by using a widely studied example.     

An adjoint method for the inverse design of solidification processes with natural convection

This paper presents a finite element algorithm based on the adjoint method for the design of a certain class of solidification processes. In particular, the paper addresses the design of directional solidification processes for pure materials such that a desired freezing front heat flux and growth velocity are achieved. This is the first time that an infinite-dimensional continuum adjoint formulation is obtained and implemented for the solution of such inverse/design problems with moving boundaries and Boussinesq incompressible flow. The present design problem belongs to a category of inverse problems in which one is looking for the unknown conditions in part of the boundary, while overspecified boundary conditions are supplied in another part of the boundary (here the freezing interface). The solidification design problem is mathematically posed as a whole time-domain optimization problem. The gradient of the cost functional is calculated using the solution of an appropriately defined continuous adjoint problem. The minimization process is realized by the conjugate gradient method via the solutions of the direct, adjoint and sensitivity sub-problems. The proposed methodology is demonstrated with the solidification of an initially superheated liquid aluminum confined in a square mold. The non-uniformity in the casting product in the direction of gravity due to the existence of natural convection in the melt is emphasized. The inverse design problem is then posed as finding the appropriate spatial-temporal variations of the boundary heat flux on the vertical mold walls that can eliminate or reduce the effects of convection on the freezing interface heat fluxes and growth velocity. The numerical example demonstrates the accuracy and convergence of the adjoint formulation. Finally, open related research design problems are discussed. © 1998 John Wiley & Sons, Ltd.     

FINITE ELEMENT SOLUTION OF A MODEL FREE SURFACE PROBLEM BY THE OPTIMAL SHAPE DESIGN APPROACH

Optimal shape design approach is applied to numerical computation of a model potential free boundary value problem. The problem is discretized using the finite element method. To test the approach the problem is formulated in both velocity potential and stream function formulation and four different finite element discretizations are used. Associated minimization problem is solved using the quasi-Newton method. Gradient of the cost function is computed by solving the algebraic adjoint equation. Gravity and surface tension forces are included in the model. Viability of the method is showed by solving problems with important effects of gravity and surface tension forces. © 1997 by John Wiley & Sons, Ltd.     

An alternative approach to shape design sensitivity analysis

A novel method is presented in this paper for calculating shape design sensitivity, which is based on the finite difference method (FDM). By analysing the numerical procedure of the FDM, the perturbation of the geometry is replaced by a perturbation load which can be calculated once the stress field of the initial problem and the design boundary perturbation are known. The final shape design sensitivity is obtained by solving the perturbation problem which has the same geometry and the kinematical boundary condition as the initial problem, but under the perturbation loads. Therefore the new method does not require the calculation of the matrices of the perturbed structure, and is independent of the perturbation step. A numerical implementation of the finite difference load method (FDLM) is described in which the boundary element method is used to evaluate the structural response. The numerical examples demonstrate that this new method for shape design sensitivity analysis is very accurate.     

SHAPE OPTIMIZATION OF A 2D BODY SUBJECTED TO SEVERAL LOADING CONDITIONS

An algorithm for shape optimization based on simultaneous solution of the equations and inequalities arising from Kuhn-Tucker necessary conditions is presented. Regular triangular FE assembly is proposed. Element vertices are associated with design variables directly or through spline parameters defining the boundary of the optimized body. This way, during the iteration procedure, FE assembly is automatically remeshed together with the motion of the optimized boundary. Multiple loading conditions are represented in the problem as equality conditions in the form of a set of equilibrium equations for each loading condition separately. From the necessary condition equations an additional, important relation between cost function, Lagrange multipliers associated with inequality constraints and their limit values is derived. The algorithm combines standard professional FEM programs with an optimizer proposed in the paper which is illustrated with shape optimization of several 2D bodies. The proposed approach is theoretically rigorous and relatively simple for practical applications, and allows considerations sensitivities, adjoint systems and constraints linearization to be avoided.     

Shape design sensitivity analysis and optimization of two-dimensional periodic thermal problems using BEM

 A general procedure to perform shape design sensitivity analysis for two-dimensional periodic thermal diffusion problems is developed using boundary integral equation formulation. The material derivative concept to describe shape variation is used. The temperature is decomposed into a steady state component and a perturbation component. The adjoint variable method is used by utilizing integral identities for each component. The primal and adjoint systems are solved by boundary element method. The sensitivity results compared with those by finite difference show good accuracy. The shape optimal design problem of a plunger model for the panel of a television bulb, which operates periodically, is solved as an example. Different objectives and amounts of heat flux allowed are studied. Corresponding optimum shapes of the cooling boundary of the plunger are obtained and discussed. Received 15 August 2001 / Accepted 28 February 2002     

Boundary integral equation method for shape optimization of elastic structures

A general method for shape design sensitivity analysis as applied to plane elasticity problems is developed with a direct boundary integral equation formulation, using the material derivative concept and adjoint variable method. The problem formulation is very general and a complete consideration is given to describing the boundary variation by including the tangential component of the velocity field. The method is then applied to obtain the sensitivity formula for a general stress constraint imposed over a small part of the boundary. The accuracy of the design sensitivity analysis is studied with a fillet and an elastic ring design problem. Among the several numerical implementations tested, the second order boundary elements with a cubic spline representation of the moving boundary have shown the best accuracy. A smooth characteristic function is found to be better than a plateau function for localization of the stress constraint. Optimal shapes for the two problems are presented to show numerical applications.     

Shape optimization using the boundary element method with substructuring

Applications of the boundary element method for two- and three-dimensional structural shape optimization are presented. The displacements and stresses are computed using the boundary element method. Sub-structuring is used to isolate the portion of the structure undergoing geometric change. The corresponding non-linear programming problem for the optimization is solved by the generalized reduced gradient method. B-spline curves and surfaces are introduced to describe the shape of the design. The control points on these curves or surfaces are selected as design variables. The design objective may be either to minimize the weight or a peak stress of the component by determining the optimum shape subject to geometrical and stress constraints. The use of substructuring allows for problem solution without requiring traditional simplifications such as linearization of the constraints. The method has been successfully applied to the structural shape optimization of plane stress, plane strain and three-dimensional elasticity problems.     

Adjoint shape design sensitivity analysis of molecular dynamics for lattice structures using GLE impedance forces

We presented a shape design sensitivity analysis method for lattice structures using a generalized Langevin equation (GLE) to overcome the difficulty of discrete nature in atomic systems. Taking advantage of the GLE forces, the perturbed atomistic region is treated as the GLE impedance forces and the shape design problem of discrete atomic variations is converted into a non-shape problem with GLE impedance forces. We developed an adjoint variable method in order to improve the computational efficiency for molecular dynamics (MD) with many design variables. Due to the translational symmetry in lattice structure, the size of the time history kernel function that accounts for the boundary effects of reduced systems could be reduced to that of a single atoms DOFs. In numerical examples, the convergent characteristic of shape sensitivity according to the amount of shape variations is investigated in MD systems. Also, the results of the derived shape sensitivity turn out to be more accurate and efficient, compared with those of the finite difference ones.     

Entropy minimization in turbine cascade using continuous adjoint formulation

A complete continuous adjoint formulation is presented here for the optimization of the turbulent flow entropy generation rate through a turbine cascade. The adjoint method allows one to have many design variables, but still afford to compute the objective function gradient. The new adjoint system can be applied to different structured and unstructured grids as well as mixed subsonic and supersonic flows. For turbulent flow simulation, the k–ω shear-stress transport turbulence model and Roe's flux function are used. To ensure all possible shape models, a mesh-point method is used for design parameters, and an implicit smoothing function is implemented to avoid the generation of non-smoothed blades. To analyse the capability of the presented algorithm, the shape of a turbine cascade blade is redesigned and a few physical observations are made on how the scheme improves the blade performance.     

Adjoint design sensitivity analysis of molecular dynamics in parallel computing environment

An adjoint design sensitivity analysis method is developed for molecular dynamics using a parallel computing scheme of spatial decomposition in both response and design sensitivity analyses to enhance the computational efficiency. Molecular dynamics is a path-dependent transient dynamic problem with many design variables of high nonlinearity. Adjoint variable method is not appropriate for path-dependent problems but employed in this paper since the path is readily available from response analysis. The required adjoint system is derived as a terminal value problem. To compute the interaction forces between atoms in different spatial boxes, only atomic positions in the neighboring boxes are required to minimize the amount of data communications. Through some numerical examples, the high nonlinearity of the selected design variables is discussed. Also, the accuracy of the derived adjoint design sensitivity is verified by comparing with finite difference sensitivity and the efficiency of parallel adjoint variable method is demonstrated.