Using artificial reaction force to design compliant mechanism with multiple equality displacement constraints

Monolithic compliant mechanisms are elastic workpieces which transmit force and displacement from an input position to an output position. Continuum topology optimization is suitable to generate the optimized topology, shape and size of such compliant mechanisms. The optimization strategy for a single input single output compliant mechanism under volume constraint is known to be best implemented using an optimality criteria or similar mathematical programming method. In this standard form, the method appears unsuitable for the design of compliant mechanisms which are subject to multiple outputs and multiple constraints. Therefore an optimization model that is subject to multiple design constraints is required. With regard to the design problem of compliant mechanisms subject to multiple equality displacement constraints and an area constraint, we here present a unified sensitivity analysis procedure based on artificial reaction forces, in which the key idea is built upon the Lagrange multiplier method. Because the resultant sensitivity expression obtained by this procedure already compromises the effects of all the equality displacement constraints, a simple optimization method, such as the optimality criteria method, can then be used to implement an area constraint. Mesh adaptation and anisotropic filtering method are used to obtain clearly defined monolithic compliant mechanisms without obvious hinges. Numerical examples in 2D and 3D based on linear small deformation analysis are presented to illustrate the success of the method.     

Adjoint sensitivity analysis for differential-algebraic equations: algorithms and software

An efficient numerical method for sensitivity computation of large-scale differential-algebraic systems is developed based on the adjoint method. Issues that are critical for the implementation are addressed. Complexity analysis and numerical results demonstrate that the adjoint sensitivity method is advantageous over the forward sensitivity method for applications with a large number of sensitivity parameters and few objective functions.     

Adjoint based topology optimization of conjugate heat transfer systems

This paper presents a topology optimization method for coupled thermal problems. Heat transfer linked with the forced convection flow inside cooling channels is investigated using a conjugate model. This model includes both the full Navier-Stokes equations for the fluid medium and the energy equations for both fluid and solid. In this present work, the adjoint method is extended to such conjugate heat transfer (CHT) systems to optimize their performance by the use of gradient based methods. This performance is usually a compromise between an increase in heat flux or temperature distribution at a surface and maintaining a low pressure loss within the system. To exemplify the method a uniform temperature distribution is chosen and evaluated numerically. For implementation the open source CFD Software OpenFOAM is used. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Efficient Aerodynamic Shape Optimization in MDO Context

The aerospace industry is increasingly relying on advanced numerical flow simulation tools in the early aircraft design phase. Today's flow solvers, which are based on the solution of the compressible Euler and Navier-Stokes equations, are able to predict aerodynamic behaviour of aircraft components under different flow conditions quite well [1]. Within the next few years numerical shape optimization will play a strategic role for future aircraft design. It offers the possibility of designing or improving aircraft components with respect to a pre-specified figure of merit, subject to geometrical and physical constraints. Here, aero-structural analysis is necessary to reach physically meaningful optimum wing designs. The use of single disciplinary optimizations applied in sequence is not only inefficient but in some cases is known to lead to wrong, non-optimal designs [2]. Although multidisciplinary optimizations (MDO) are possible with classical approaches for sensitivity evaluations by means of finite differences, these methods are extremely expensive in terms of calculation time, requiring the reiterated solution of the coupled problem for every design variable. However, adjoint approaches allow the evaluation of these sensitivities in an efficient way and lead to high accuracy. Firstly, we present the development and application of a continuous adjoint approach for single disciplinary aerodynamic shape design. This approach was previously developed at the German Aerospace Center (DLR) [3] and was the starting point for the extension to aero-structural wing designs. Secondly, we describe the adjoint approach and its implementation for the evaluation of the sensitivities for coupled aero-structure optimization problems [4] and its application to the drag reduction of the AMP wing by constant lift while taking into account the static deformation of this wing caused by the aerodynamic forces (see figures). Finally, we show the application of the coupled aero-structural adjoint approach for the Breguet formula of aircraft range, where in addition to the lift to drag ratio the weight of the AMP wing is taken into account (see figures). (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Efficient aerodynamic shape optimization in MDO context

Multidisciplinary analysis is necessary to reach physically meaningful optimum designs. For aero-structural shape optimization this means coupling two disciplines—aerodynamics and structural mechanics. In this paper, the sensitivity evaluation for aerodynamic shape optimization is considered, while taking into account the static aeroelastic effects introduced by the variations in the aerodynamic forces, which are associated with changes in the aerodynamic shape. Due to the high computational cost of a finite difference evaluation step for such a coupled problem, an extension of the adjoint approach to aeroelasticity is necessary for an efficient calculation of the sensitivities. The implementation, validation and application of such a method in the MDO context described above are presented.     

Evolutionary structural optimization for problems with stiffness constraints

This paper presents a simple evolutionary procedure based on finite element analysis to minimize the weight of structures while satisfying stiffness requirements. At the end of each finite element analysis, a sensitivity number, indicating the change in the stiffness due to removal of each element, is calculated and elements which make the least change in the stiffness; of a structure are subsequently removed from the structure. The final design of a structure may have its weight significantly reduced while the displacements at prescribed locations are kept within the given limits. The proposed method is capable of performing simultaneous shape and topology optimization. A wide range of problems including those with multiple displacement constraints, multiple load cases and moving loads are considered. It is shown that existing solutions of structural optimization with stiffness constraints can easily be reproduced by this proposed simple method. In addition some original shape and layout optimization results are presented.     

An Improved Hybrid Adjoint Method in External Aerodynamics Using Variational Technique for the Boundary Integral Based Optimal Objective Function Gradient

An improved hybrid adjoint method to the viscous, compressible Reynold-Averaged Navier-Stokes Equation (RANS) is developed for the computation of objective function gradient and demonstrated for external aerodynamic design optimization. In this paper, the main idea is to extend the previous coupling of the discrete and continuous adjoint method by the grid-node coordinates variation technique for the computation of the variation in the gradients of flow variables. This approach in combination with the Jacobian matrices of flow fluxes refrained the objective function from field integrals and coordinate transformation matrix. Thus, it opens up the possibility of employing the hybrid adjoint method to evaluate the subsequent objective function gradient analogous to many shape parameters, comprises of only boundary integrals. This avoids the grid regeneration in the geometry for every surface perturbation in a structured and unstructured grid. Hence, this viable technique reduces the overall CPU cost. Moreover, the new hybrid adjoint method has been successfully applied to the computation of accurate sensitivity derivatives. Finally, for the investigation of the presented numerical method, simulations are carried out on NACA0012 airfoil in a transonic regime and its accuracy and effectiveness related to the new gradient equation have been verified with the Finite Difference Method (FDM). The analysis reveals that the presented methodology for the optimization provides the designer with an indispensable CPU-cost effective tool to reshape the complex geometry airfoil surfaces, useful relative to the state-of-the-art, in a less computing time.     

A study of design velocity field computation for shape optimal design

Design velocity field computation is an important step in computing shape design sensitivity coefficients and updating a finite element mesh in the shape design optimization process. Applying an inappropriate design velocity field for shape design sensitivity analysis and optimization will yield inaccurate sensitivity results or a distorted finite element mesh, and thus fail in achieving an optimal solution. In this paper, theoretical regularity and practical requirements of the design velocity field are discussed. The crucial step of using the design velocity field to update the finite element mesh in the design optimization process is emphasized. Available design velocity field computation methods in the literature are summarized and their applicability for shape design sensitivity analysis and optimization is discussed. Five examples are employed to discuss applicability of these methods. It was found that a combination of isoparametric mapping and boundary displacement methods is ideal for the design velocity field computation.     

Bi-directional evolutionary topology optimization of geometrically nonlinear continuum structures with stress constraints

This paper proposes a design method to maximize the stiffness of geometrically nonlinear continuum structures subject to volume fraction and maximum von Mises stress constraints. An extended bi-directional evolutionary structural optimization (BESO) method is adopted in this paper. BESO method based on discrete variables can effectively avoid the well-known singularity problem in density-based methods with low density elements. The maximum von Mises stress is approximated by the p-norm global stress. By introducing one Lagrange multiplier, the objective of the traditional stiffness design is augmented with p-norm stress. The stiffness and p-norm stress are considered simultaneously by the Lagrange multiplier method. A heuristic method for determining the Lagrange multiplier is proposed in order to effectively constrain the structural maximum von Mises stress. The sensitivity information for designing variable updates is derived in detail by adjoint method. As for the highly nonlinear stress behavior, the updated scheme takes advantages from two filters respectively of the sensitivity and topology variables to improve convergence. Moreover, the filtered sensitivity numbers are combined with their historical sensitivity information to further stabilize the optimization process. The effectiveness of the proposed method is demonstrated by several benchmark design problems.     

Hybrid evolutionary algorithm with Hermite radial basis function interpolants for computationally expensive adjoint solvers

In this paper, we present an evolutionary algorithm hybridized with a gradient-based optimization technique in the spirit of Lamarckian learning for efficient design optimization. In order to expedite gradient search, we employ local surrogate models that approximate the outputs of a computationally expensive Euler solver. Our focus is on the case when an adjoint Euler solver is available for efficiently computing the sensitivities of the outputs with respect to the design variables. We propose the idea of using Hermite interpolation to construct gradient-enhanced radial basis function networks that incorporate sensitivity data provided by the adjoint Euler solver. Further, we conduct local search using a trust-region framework that interleaves gradient-enhanced surrogate models with the computationally expensive adjoint Euler solver. This ensures that the present hybrid evolutionary algorithm inherits the convergence properties of the classical trust-region approach. We present numerical results for airfoil aerodynamic design optimization problems to show that the proposed algorithm converges to good designs on a limited computational budget.     

Topology Optimization of Flexible Multibody Systems Using Equivalent Static Loads and Displacement Fields

Topology optimization techniques are applied in most cases for static applications. However, recently topology optimization procedures for structures under dynamic loads have been the focus of several studies. In this work, a topology optimization scheme for flexible multibody systems using equivalent static loads and displacement fields is investigated. The optimization problem is formulated using a homogenization method, more precisely, the solid isotropic material with penalization (SIMP) approach. The objective function in the optimization problem is the compliance and the method of moving asymptotes is used as optimizer. The objective function and the sensitivities are computed directly from the displacement field computed in the dynamic simulation. The examples of a 2-arm manipulator and a slider-crank mechanism are presented and the results are discussed to verify the improved dynamical behavior through this optimization method. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A problem of generalized magneto-thermoelastic thin slim strip subjected to a moving heat source

The generalized thermoelastic theory with thermal relaxation, in the context of Lord and Shulman theory, is used to investigate the magneto-thermoelastic problem of a thin slim strip placed in a magnetic field and subjected to a moving plane of heat source. The generalized magneto-thermoelastic coupled governing equations are formulated. By means of the Laplace transform and numerical Laplace inversion, the governing equations are solved. Numerical calculations for the considered variables are performed and the obtained results are presented graphically. The effects of moving heat source speed and applied magnetic field on temperature, stress and displacement are studied. It is found from the graphs that the temperature, thermally induced displacement and stress in the strip are found to decrease at large heat source speed, and the magnetic field significantly influences the variations of non-dimensional displacement and stress. However, it has no effect on the non-dimensional temperature.     

应力和位移约束下连续体结构拓扑优化

同时考滤应力和位移约束的连续体结构拓扑优化问题,很难用现有的均匀方法或变密度方法等求解。主要困难在于难以建立应力和位移约束与拓扑设计变量间显式关系式;即使建立了这种关系,也由于优化问题规模过大,利用常规的数学规划方法难以求解。隋允康、杨德庆曾提出了基于独立连续拓扑变量及映射变换（ＩＣＭ）的桁架结构拓扑优化模型。本文在此基础上,建立了以重量为目标,考虑应力和位移约束的连续体结构拓扑优化模型,并推导出     

Robust topology optimization for multi-material structures under interval uncertainty

In this paper, we propose an efficient method to design robust multi-material structures under interval loading uncertainty. The objective of this study is to minimize the structural compliance of linear elastic structures. First, the loading uncertainty can be decomposed into two unit forces in the horizontal and vertical directions based on the orthogonal decomposition, which separates the uncertainty into the calculation coefficients of structural compliance that are not related to the finite element analysis. In this manner, the time-consuming procedure, namely, the nested double-loop optimization, can be avoided. Second, the uncertainty problem can be transformed into an augmented deterministic problem by means of uniform sampling, which exploits the coefficients related to interval variables. Finally, an efficient sensitivity analysis method is explicitly developed. Thus, the robust topology optimization (RTO) problem considering interval uncertainty can be solved by combining orthogonal decomposition with uniform sampling (ODUS). In order to eliminate the influence of numerical units when comparing the optimal results to deterministic and RTO solutions, the relative uncertainty related to interval objective function is employed to characterize the structural robustness. Several multi-material structure optimization cases are provided to demonstrate the feasibility and efficiency of the proposed method, where the magnitude uncertainty, directional uncertainty, and combined uncertainty are investigated.     

轻质热防护系统波纹夹芯结构热力耦合分析

高超声速飞行器在出入大气层或持续在空间飞行时,将遭受严苛的气动加热载荷．对热防护系统进行传热分析是进行热力耦合分析的基础,而温度分布的特点直接影响到波纹夹芯结构的热应力等问题．首先对一体化热防护系统（integrated thermal protection system, ITPS）进行隔热性能分析,得到整个结构的温度场;然后采用顺序耦合的数值方法,模拟分析ITPS波纹夹芯结构单胞的热力耦合性能,给出波纹夹芯结构在静力载荷以及热力耦合载荷条件下的应力场、位移场,并对计算结果进行了讨论．结果表明波纹夹芯结构在初始尺寸及约束条件下,只满足在高温热流作用下飞行器低压区使用,而当气动压力大于等于15 000 Pa时,结构将发生破坏．     

n-sided polygonal hybrid finite elements with unified fundamental solution kernels for topology optimization

In topology optimization, the optimized design can be obtained based on spatial discretization of design domain using natural polygonal finite elements to reduce the influence of mesh geometry on topology optimization solutions. However, the natural polygonal finite elements require separate interpolants for each type of elements and involve troublesome domain integrals. In this study, an alternative n-sided polygonal hybrid finite element possessing multiple-node connection is formulated in a unified form to compress the checkerboard patterns caused by numerical instability in topology optimization. Different from the natural polygonal finite elements, the present polygonal hybrid finite elements involve two sets of independent displacement fields. The intra-element displacement field defined inside the element is approximated by the linear combination of the fundamental solution of the problem to achieve the purpose of the local satisfaction of the governing equations of the problem, but not the specific boundary conditions and the inter-element continuity conditions. To overcome such drawback, the inter-element displacement field defined over the entire element boundary is independently approximated by means of the conventional shape function interpolation. As a result, only line integrals along the element boundary are involved in the computation, whose dimension is reduced by one compared to the domain integrals in the natural polygonal finite elements, and more importantly, allowing us to flexibly construct any polygons from Voronoi tessellations in discretizing complex design domains using same fundamental solution kernels. Numerical results obtained indicate that the present n-sided polygonal hybrid finite elements can produce more accurate displacement solutions and smaller mean compliance, compared to the standard finite elements and the natural polygonal finite elements.     

Shape Optimal Design Problem with Convective and Radiative Heat Transfer: Analysis and Implementation

We present a study of an optimal design problem for a coupled system, governed by a steady-state potential flow equation and a thermal equation taking into account radiative phenomena with multiple reflections. The state equation is a nonlinear integro-differential system. We seek to minimize a cost function, depending on the temperature, with respect to the domain of the equations. First, we consider an optimal design problem in an abstract framework and, with the help of the classical adjoint state method, give an expression of the cost function differential. Then, we apply this result in the two-dimensional case to the nonlinear integro-differential system considered. We prove the differentiability of the cost function, introduce the adjoint state equation, and give an expression of its exact differential. Then, we discretize the equations by a finite-element method and use a gradient-type algorithm to decrease the cost function. We present numerical results as applied to the automotive industry.     

An efficient sensitivity computation strategy for the evolutionary structural optimization (ESO) of continuum structures subjected to self-weight loads

This work presents a modified version of the evolutionary structural optimization procedure for topology optimization of continuum structures subjected to self-weight forces. Here we present an extension of this procedure to deal with maximum stiffness topology optimization of structures when different combinations of body forces and fixed loads are applied. Body forces depend on the density distribution over the design domain. Therefore, the value and direction of the loading are coupled to the shape of the structure and they change as the material layout of the structure is modified in the course of the optimization process. It will be shown that the traditional calculation of the sensitivity number used in the ESO procedure does not lead to the optimum solution. Therefore, it is necessary to correct the computation of the element sensitivity numbers in order to achieve the optimum design. This paper proposes an original correction factor to compute the sensitivities and enhance the convergence of the algorithm. The procedure has been implemented into a general optimization software and tested in several numerical applications and benchmark examples to illustrate and validate the approach, and satisfactorily applied to the solution of 2D, 3D and shell structures, considering self-weight load conditions. Solutions obtained with this method compare favourably with the results derived using the SIMP interpolation scheme.     

Optimality criteria coupled adaptive mesh method for optimal shape design of Stokes flow

An adaptive mesh method combined with the optimality criteria algorithm is applied to optimal shape design problems of fluid dynamics. The shape sensitivity analysis of the cost functional is derived. The optimization problem is solved by a simple but robust optimality criteria algorithm, and an automatic local adaptive mesh refinement method is proposed. The mesh adaptation, with an indicator based on the material distribution information, is itself shown as a shape or topology optimization problem. Taking advantages of this algorithm, the optimal shape design problem concerning fluid flow can be solved with higher resolution of the interface and a minimum of additional expense. Details on the optimization procedure are provided. Numerical results for two benchmark topology optimization problems are provided and compared with those obtained by other methods. Copyright © 2016 John Wiley & Sons, Ltd.