Numerical Methods for Accurate Computation of Design Sensitivities for Two-Dimensional Navier-Stokes Problems

Gradient computations can be a limiting factor in algorithm efficiency and accuracy for optimization based design. In this paper, we present three parameterized flow problems and consider the evaluation of state sensitivities both theoretically and numerically. Existence and uniqueness results are given for the sensitivities of a specific group of two-dimensional Navier-Stokes problems. We then turn our attention to obtaining numerical approximations to state sensitivities. We show convergence of our numerical sensitivities using a problem having an exact solution. Next, two problems, flow around a cylinder and flow over a bump, are used to evaluate several computational schemes. In particular, a local projection scheme for improved state derivative approximations and the use of an adaptive finite element scheme are shown to be important techniques for obtaining accurate sensitivity approximations. Lastly, we evaluate the impact of these computational techniques on cost function and gradient calculation.     

Eigenfrequency optimization of stiffened plate using Kriging estimation

This paper describes the optimization of the beam stiffeners attached to plates in an eigenfrequency problem. The solution space is estimated using the Kriging method. A finite element analysis is carried out to evaluate the objective function at the sample points used for the estimation. The gradient method is used as a local optimizer. The Kriging estimation incurs relatively low cost, and is easy to combine with the gradient method. In this paper, we solve eigenfrequency optimization problems for a fully supported plate to maximize the minimum eigenfrequency and the difference between the 1st- and 2nd-order eigenfrequency. An optimization problem for an L-shaped plate with an eigenfrequency constraint is also solved. Good solutions are obtained for each example, and all the optimizations for these problems can be done at a lower computational cost. The results highlight the effectiveness of the method to solve the eigenfrequency optimization problems for the stiffened plate.     

A multiobjective topology optimization approach for cost and time minimization in additive manufacturing

The ever-present drive for increasingly high-performance designs realized on shorter timelines has fostered the need for computational design generation tools such as topology optimization. However, topology optimization has always posed the challenge of generating difficult, if not impossible to manufacture designs. The recent proliferation of additive manufacturing technologies provides a solution to this challenge. The integration of these technologies undoubtedly has the potential for significant impact in the world of mechanical design and engineering. This work presents a new methodology which mathematically considers additive manufacturing cost and build time alongside the structural performance of a component during the topology optimization procedure. Two geometric factors, namely, the surface area and support volume required for the design, are found to correlate to cost and build time and are controlled through the topology optimization procedure. A novel methodology to consider each of these factors dynamically during the topology optimization procedure is presented. The methodology, based largely on the use of the spatial gradient of the density field, is developed in such a way that it does not leverage the finite element discretization scheme. This work investigates a problem that has not yet been explored in the literature: direct minimization of support material volume in density-based topology optimization. The entire methodology is formulated in a smooth and differentiable manner, and the sensitivity expressions required by gradient based optimization solvers are presented. A series of example problems are provided to demonstrate the efficacy of the proposed methodology.     

SIMULTANEOUS OPTIMIZATION OF SHAPE AND FLOW PARAMETERS FOR COMBINED FREE AND FORCED CONVECTION IN VERTICAL CHANNELS

Simultaneous optimization of shape and flow parameters is performed for a combined free and forced convection flow through vertical rectangular channels with moving walls. The laminar flow is assumed to be fully developed in the axial direction. The wall velocity, the axial pressure gradient and the channel height in the transverse plane are taken as the optimization parameters. The sensitivity expressions of both the objective function and the flow rate constraint of optimization are obtained in terms of the relevant physical variables, as well as adjoint variables which satisfy additional p.d.e.'s. All equations are discretized using the finite element method. Numerical results are provided for the present constrained optimization problem for various values of the problem parameters which include the moving wall segment size and the Rayleigh number. The results indicate that with increased Rayleigh number the optimal values of the wall velocity and the axial pressure gradient are increased, while the optimal value of the channel height is decreased. General sensitivity expressions are also presented in the appendix which might be utilized for arbitrary boundary variations along with arbitrary optimization objectives in other investigations. © 1997 by 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 algorithm for optimizing CVBEM and BEM nodal point locations

The Complex Variable Boundary Element Method or CVBEM is a numerical technique for approximating particular partial differential equations such as the Laplace or Poisson equations (which frequently occur in physics and engineering problems, among many other fields of study). The advantage in using the CVBEM over traditional domain methods such as finite difference or finite element based methods includes the properties that the resulting CVBEM approximation is a function: (i) defined throughout the entire plane, (ii) that is analytic throughout the problem domain and almost everywhere on the problem boundary and exterior of the problem domain union boundary; (iii) is composed of conjugate two-dimensional real variable functions that are both solutions to the Laplace equation and are orthogonal such as to provide the “flow net” of potential and stream functions, among many other features. In this paper, a procedure is advanced that locates CVBEM nodal point locations on and exterior of the problem boundary such that error in matching problem boundary conditions is reduced. That is, locating the nodal points is part of modeling optimization process, where nodes are not restricted to be located on the problem boundary (as is the typical case) but instead locations are optimized throughout the exterior of the problem domain as part of the modeling procedure. The presented procedure results in nodal locations that achieve considerable error reduction over the usual methods of placing nodes on the problem boundary such as at equally spaced locations or other such procedures. Because of the significant error reduction observed, the number of nodes needed in the model is significantly reduced. It is noted that similar results occur with the real variable boundary element method (or BEM).The CVBEM and relevant nodal location optimization algorithm is programmed to run on program Mathematica, which provides extensive internal modeling and output graphing capabilities, and considerable levels of computational accuracy. The Mathematica source code is provided.     

An inverse method for material parameters estimation in the inelastic range

Inverse problem theory is getting of an increasing importance in mechanical modelling as it brings a solution to the identification to rheological behaviour of materials in the nonlinear range. As a matter of fact, when using inverse identification, the problem of experimental tests interpretation associated to inhomogeneous deformation states is bypassed. This allows a more accurate material parameters determination compared to the direct identification. In this paper, an inverse identification method is proposed to determine material parameters in the nonlinear range. The algorithm developed consists of a finite element based inversion scheme associated to an optimization procedure. A sensitivity analysis is used in order to determine the gradient of the cost function, representing the difference between the measured and the calculated response, with respect to the material parameters to identify. This method is applied to the inverse identification of viscoplastic parameters entering in the constitutive function that describes the flow stress of an aluminium alloy, for large range of strain, strain rate and temperature.     

Multidisciplinary Design Optimization Procedure for Improved Design of a Cooled Gas Turbine Blade

A multidisciplinary optimization procedure for gas turbine blade design has been developed and demonstrated on a generic 3-D blade. The blade is cooled both internally and externally (film cooling). Aerodynamic and heat transfer design criteria are integrated along with various constraints on the blade geometry. The blade is divided into numerous spanwise sections and each section is represented by a Bezier-Bernstein polynomial. A comprehensive solver for 3-D Navier-Stokes equations is used for the viscous flow calculations. The finite element method is used to obtain the blade interior temperatures. The average blade temperature and maximum blade temperature at each spanwise section are minimized, with aerodynamic and geometric constraints on the blade geometry. The constrained multiobjective optimization problem is solved using the Kreisselmeier-Steinhauser function approach. The results for a generic turbine blade design problem show significant improvements after optimization.     

Inverse design of directional solidification processes in the presence of a strong external magnetic field

A computational method for the design of directional alloy solidification processes is addressed such that a desired growth velocity ν_f under stable growth conditions is achieved. An externally imposed magnetic field is introduced to facilitate the design process and to reduce macrosegregation by the damping of melt flow. The design problem is posed as a functional optimization problem. The unknowns of the design problem are the thermal boundary conditions. The cost functional is taken as the square of the L₂ norm of an expression representing the deviation of the freezing interface thermal conditions from the conditions corresponding to local thermodynamic equilibrium. The adjoint method for the inverse design of continuum processes is adopted in this work. A continuum adjoint system is derived to calculate the adjoint temperature, concentration, velocity and electric potential fields such that the gradient of the L₂ cost functional can be expressed analytically. The cost functional minimization process is realized by the conjugate gradient method via the FE solutions of the continuum direct, sensitivity and adjoint problems. The developed formulation is demonstrated with an example of designing the boundary thermal fluxes for the directional growth of a germanium melt with dopant impurities in the presence of an externally applied magnetic field. The design is shown to achieve a stable interface growth at a prescribed desired growth rate. Copyright © 2001 John Wiley & Sons, Ltd.     

Simulation and optimization of heat flow via anisotropic material thermal conductivity

This article is focused on heat flow control within a composite material that has designed anisotropic thermal conductivity. The optimal conductive heat transfer path in the composite is specified via customized local scale properties, where the physical parameter distribution is found using an iterative procedure that couples a gradient based optimization routine with a finite element solver. A sample optimization result is presented to illustrate the procedure, and the final solution is translated into a physical embodiment having heterogeneous material properties. Numerical experiments were performed both on this synthesized material and a baseline homogeneous material with the same filler volume fraction. Heat transfer results indicate a substantial reduction in overall temperature with effective concentration of thermal power density in the designed material.     

A gradient‐only line search method for the conjugate gradient method applied to constrained optimization problems with severe noise in the objective function

A new implementation of the conjugate gradient method is presented that economically overcomes the problem of severe numerical noise superimposed on an otherwise smooth underlying objective function of a constrained optimization problem. This is done by the use of a novel gradient‐only line search technique, which requires only two gradient vector evaluations per search direction and no explicit function evaluations. The use of this line search technique is not restricted to the conjugate gradient method but may be applied to any line search descent method. This method, in which the gradients may be computed by central finite differences with relatively large perturbations, allows for the effective smoothing out of any numerical noise present in the objective function. This new implementation of the conjugate gradient method, referred to as the ETOPC algorithm, is tested using a large number of well‐known test problems. For initial tests with no noise introduced in the objective functions, and with high accuracy requirements set, it is found that the proposed new conjugate gradient implementation is as robust and reliable as traditional first‐order penalty function methods. With the introduction of severe relative random noise in the objective function, the results are surprisingly good, with accuracies obtained that are more than sufficient compared to that required for engineering design optimization problems with similar noise. Copyright © 2004 John Wiley & Sons, Ltd.     

Optimization of product and process design for quality and cost

Robust product and process design is an important technique for achieving high quality at low cost. It involves making the product's function much less sensitive to various sources of noise such as manufacturing variation, environmental variation and deterioration. This is a problem in optimization involving minimization of the mean square loss resulting from the deviation of the product's function from its target. Here we show that the optimization can be carried out in two steps: first maximize a quantity called signal-to-noise ratio (S/N) and then bring the performance on target by special adjustment parameters. The two-step procedure works for a wide variety of product functions and makes the optimization process more efficient and practical compared to the direct minimization of the quadratic loss function.     

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.     

采用等效有限元模型的复合材料机翼结构优化

在机翼设计过程中,将等效有限元模型(EFEM)方法应用于考虑静力学和动力学要求的机翼结构优化。提出了"三步走"的结构优化策略,将一个多变量的复杂优化问题转换为一系列少变量的简单优化问题,对某支线客机的复合材料机翼进行了优化设计。首先以位移、静强度和颤振速度作为约束条件对机翼复合材料铺层比例进行优化;然后以静强度和结构稳定性作为约束,以最小化结构质量和结构效率作为优化目标,对各翼肋之间的加强壁板进行优化设计;最后再以位移和颤振速度为约束,对机翼结构总体刚度进行优化设计。结果表明:EFEM方法具有快速建模和计算量少的优点,采用"三步走"优化策略具有更高的效率,适用于初步机翼结构优化设计。     

基于密度分布曲线法的复合材料变刚度铺层优化

基于梯度的优化方法对复合材料层合板进行了变刚度铺层优化设计。在优化过程中需确定铺层中各单元的密度以及角度。为了使优化结果具有可制造性,优化结果需满足制造工艺约束并且铺层角度需从预定角度中选取。为了避免在优化问题中引入过多的约束并减少设计变量的数目,提出密度分布曲线法(DDCM)对层合板中各单元的密度进行参数化。根据各单元的密度以及角度设计变量并基于Bi-value Coding Parameterization(BCP)方法中的插值公式确定各单元的弹性矩阵。优化过程中以结构柔顺度作为优化目标,结构体积作为约束,优化算法采用凸规划对偶算法。对碳纤维复合材料的算例结果表明:采用DDCM可得到较理想的优化结果,并且收敛速率较快。     

An integer concave minimization approach for the minimum concave cost capacitated flow problem on networks

Formulating the minimum concave cost capacitated network flow problem as an integer concave minimization problem, we establish finite branch and bound algorithms, in which the branching operation is the so–called integral rectangular partition and the bounding procedure is performed by the classical minimum linear cost flow problem on subnetworks. For the special case that the flow cost function is concave on a fixed number of arcs and linear on the others, an upper bound of the running time is given. Received: 19 July 1996 / Accepted: 8 July 1997     

Optimal design of gating systems by gradient search methods

A numerical optimization technique based on gradient-search is applied to obtain an optimal design of a typical gating system used for the gravity process to produce aluminum parts. This represents a novel application of coupling nonlinear optimization techniques with a foundry process simulator, and it is motivated by the fact that a scientifically guided search for better designs based on techniques that take into account the mathematical structure of the problem is preferred to commonly found trial-and-error approaches. The simulator applies the finite volume method and the VOF algorithm for CFD analysis. The direct gradient optimization algorithm, sequential quadratic programming (SQP), was used to solve both a 2D and a 3D gating system design problems using two design variables. The results clearly show the effectiveness of the proposed approach for finding high quality castings when compared with current industry practices.     

Improving multiresolution topology optimization via multiple discretizations

Unlike the traditional topology optimization approach that uses the same discretization for finite element analysis and design optimization, this paper proposes a framework for improving multiresolution topology optimization (iMTOP) via multiple distinct discretizations for: (1) finite elements; (2) design variables; and (3) density. This approach leads to high fidelity resolution with a relatively low computational cost. In addition, an adaptive multiresolution topology optimization (AMTOP) procedure is introduced, which consists of selective adjustment and refinement of design variable and density fields. Various two‐dimensional and three‐dimensional numerical examples demonstrate that the proposed schemes can significantly reduce computational cost in comparison to the existing element‐based approach. Copyright © 2012 John Wiley & Sons, Ltd.     

Optimization of a variable mouth acoustic horn

By using boundary shape optimization on the end part of a semi‐infinite waveguide for acoustic waves, we design transmission‐efficient interfacial devices without imposing an upper bound on the mouth diameter. The boundary element method solves the Helmholtz equation modeling the exterior wave propagation problem. A gradient‐based optimization algorithm solves the resulting least‐squares problem and the adjoint method provides the necessary gradients. The results demonstrate that there appears to be a natural limit on the optimal mouth diameter. Copyright © 2010 John Wiley & Sons, Ltd.     

Simultaneous topology and build orientation optimization for minimization of additive manufacturing cost and time

The ever-present demand for increased performance in mechanical systems, and reduced cost and manufacturing time, has led to the adoption of computational design tools and innovative manufacturing methods. One such tool is topology optimization (TO), which often produces designs that are impracticable to manufacture. However, recent developments in additive manufacturing (AM) have made production of such complex designs feasible. Therefore, integration of these technologies has the potential to innovate the design and manufacture of mechanical components. This work presents a novel mathematical methodology for multiobjective minimization of structural compliance and AM cost and time, in simultaneous build orientation and density-based TO. Component surface area and support volume were implemented in this method as the physical factors influencing AM cost and time. A new methodology was produced to approximate support volume throughout TO with variable build orientation, enabling direct minimization of support volume in the proposed optimization. The methodology allows derivation of sensitivity expressions, thereby permitting the use of efficient gradient-based optimization solvers. Three numerical examples demonstrated that the proposed methodology can efficiently produce optimum build orientations and topologies, which significantly reduce structural compliance and AM cost and time.