Optimization of unsteady incompressible Navier–Stokes flows using variational level set method

This paper presents the optimization of unsteady Navier–Stokes flows using the variational level set method. The solid–liquid interface is expressed by the level set function implicitly, and the fluid velocity is constrained to be zero in the solid domain. An optimization problem, which is constrained by the Navier–Stokes equations and a fluid volume constraint, is analyzed by the Lagrangian multiplier based adjoint approach. The corresponding continuous adjoint equations and the shape sensitivity are derived. The level set function is evolved by solving the Hamilton–Jacobian equation with the upwind finite difference method. The optimization method can be used to design channels for flows with or without body forces. The numerical examples demonstrate the feasibility and robustness of this optimization method for unsteady Navier–Stokes flows.Copyright © 2012 John Wiley & Sons, Ltd.     

An explicit level set approach for generalized shape optimization of fluids with the lattice Boltzmann method

This study is concerned with a generalized shape optimization approach for finding the geometry of fluidic devices and obstacles immersed in flows. Our approach is based on a level set representation of the fluid–solid interface and a hydrodynamic lattice Boltzmann method to predict the flow field. We present an explicit level set method that does not involve the solution of the Hamilton–Jacobi equation and allows using standard nonlinear programming methods. In contrast to previous works, the boundary conditions along the fluid–structure interface are enforced by second‐order accurate interpolation schemes, overcoming shortcomings of flow penalization methods and Brinkman formulations frequently used in topology optimization. To ensure smooth boundaries and mesh‐independent results, we introduce a simple, computationally inexpensive filtering method to regularize the level set field. Furthermore, we define box constraints for the design variables that guarantee a continuous evolution of the boundaries. The features of the proposed method are studied by two numeric examples of two‐dimensional steady‐state flow problems. Copyright © 2009 John Wiley & Sons, Ltd.     

Optimal flow control for Navier–Stokes equations: drag minimization

Optimal control and shape optimization techniques have an increasing role in Fluid Dynamics problems governed by partial differential equations (PDEs). In this paper, we consider the problem of drag minimization for a body in relative motion in a fluid by controlling the velocity through the body boundary. With this aim, we handle with an optimal control approach applied to the steady incompressible Navier–Stokes equations. We use the Lagrangian functional approach and we consider the Lagrangian multiplier method for the treatment of the Dirichlet boundary conditions, which include the control function itself. Moreover, we express the drag coefficient, which is the functional to be minimized, through the variational form of the Navier–Stokes equations. In this way, we can derive, in a straightforward manner, the adjoint and sensitivity equations associated with the optimal control problem, even in the presence of Dirichlet control functions. The problem is solved numerically by an iterative optimization procedure applied to state and adjoint PDEs which we approximate by the finite element method. Copyright © 2007 John Wiley & Sons, Ltd.     

广义变分原理的结构形状优化伴随法灵敏度分析

提出了一种利用伴随变量进行结构形状优化灵敏度分析的新方法. 基于广义变分原理， 考虑了形状优化中位移边界条件的变化对结构响应的影响. 新方法弥补了Arora 等人所提出的形状优化灵敏度分析变分原理中的缺陷，为采用伴随法进行灵敏度分析提供了 新的框架.     

A level set method for structural topology optimization with multi-constraints and multi-materials

Combining the vector level set model, the shape sensitivity analysis theory with the gradient projection technique, a level set method for topology optimization with multi-constraints and multi-materials is presented in this paper. The method implicitly describes structural material interfaces by the vector level set and achieves the optimal shape and topology through the continuous evolution of the material interfaces in the structure. In order to increase computational efficiency for a fast convergence, an appropriate nonlinear speed mapping is established in the tangential space of the active constraints. Meanwhile, in order to overcome the numerical instability of general topology optimization problems, the regularization with the mean curvature flow is utilized to maintain the interface smoothness during the optimization process. The numerical examples demonstrate that the approach possesses a good flexibility in handling topological changes and gives an interface representation in a high fidelity, compared with other methods based on explicit boundary variations in the literature. The project supported by the National Natural Science Foundation of China (59805001, 10332010) and Key Science and Technology Research Project of Ministry of Education of China (No.104060)     

A simple method for crack growth in mixed mode with X-FEM

In this paper, a new method for level set update is proposed, in the context of crack propagation modeling with the extended finite element method (X-FEM) and level sets. Compared with the existing methods, such as the resolution of the Hamilton–Jacobi equations, this new method is much simpler because it does not required complex manipulations of the level sets. This method, called the “projection” method, uses both a classical discretization of the surface of the crack (segments for 2d cracks and triangles for 3d cracks) and a level set representation of the crack. This discretization is updated with respect to the position of the new crack front. Then the level sets are re-computed using the true distance to the new crack, by an orthogonal projection of each node of the structure onto the new crack surface. Then, numerical illustrations are given on 2d and 3d academic examples. Finally, three illustrations are given on 3d industrial applications.     

比例边界有限元二阶灵敏度设计及断裂力学分析

比例边界有限元是一种只需在边界上划分网格且无需基本解的半解析方法,能有效处理应力奇异性和无边界问题.论文提出了一种比例边界有限元的二阶灵敏度分析方法,可以准确而高效地求解响应关于参数的二阶梯度.首先通过建立仅需右特征向量的哈密顿矩阵特征灵敏度分析方程,发展了一种改进的比例边界有限元一阶灵敏度分析方法;其次,进一步通过构建二阶哈密顿矩阵特征灵敏度分析方程,并对比例边界有限元系统方程进行一系列二次直接微分,提出了一种半解析形式的比例边界有限元二阶灵敏度分析方法.该方法被应用于线弹性裂纹结构的形状灵敏度分析和不确定性传播分析.最后,给出了两个数值算例验证论文方法的有效性.     

Solution of the Navier–Stokes equations in velocity–vorticity form using a Eulerian–Lagrangian boundary element method

This paper describes the Eulerian–Lagrangian boundary element model for the solution of incompressible viscous flow problems using velocity–vorticity variables. A Eulerian–Lagrangian boundary element method (ELBEM) is proposed by the combination of the Eulerian–Lagrangian method and the boundary element method (BEM). ELBEM overcomes the limitation of the traditional BEM, which is incapable of dealing with the arbitrary velocity field in advection‐dominated flow problems. The present ELBEM model involves the solution of the vorticity transport equation for vorticity whose solenoidal vorticity components are obtained iteratively by solving velocity Poisson equations involving the velocity and vorticity components. The velocity Poisson equations are solved using a boundary integral scheme and the vorticity transport equation is solved using the ELBEM. Here the results of two‐dimensional Navier–Stokes problems with low–medium Reynolds numbers in a typical cavity flow are presented and compared with a series solution and other numerical models. The ELBEM model has been found to be feasible and satisfactory. Copyright © 2000 John Wiley & Sons, Ltd.     

A hybrid Lagrangian–Eulerian particle‐level set method for numerical simulations of two‐fluid turbulent flows

A coupled Lagrangian interface‐tracking and Eulerian level set (LS) method is developed and implemented for numerical simulations of two‐fluid flows. In this method, the interface is identified based on the locations of notional particles and the geometrical information concerning the interface and fluid properties, such as density and viscosity, are obtained from the LS function. The LS function maintains a signed distance function without an auxiliary equation via the particle‐based Lagrangian re‐initialization technique. To assess the new hybrid method, numerical simulations of several ‘standard interface‐moving’ problems and two‐fluid laminar and turbulent flows are conducted. The numerical results are evaluated by monitoring the mass conservation, the turbulence energy spectral density function and the consistency between Eulerian and Lagrangian components. The results of our analysis indicate that the hybrid particle‐level set method can handle interfaces with complex shape change, and can accurately predict the interface values without any significant (unphysical) mass loss or gain, even in a turbulent flow. The results obtained for isotropic turbulence by the new particle‐level set method are validated by comparison with those obtained by the ‘zero Mach number’, variable‐density method. For the cases with small thermal/mass diffusivity, both methods are found to generate similar results. Analysis of the vorticity and energy equations indicates that the destabilization effect of turbulence and the stability effect of surface tension on the interface motion are strongly dependent on the density and viscosity ratios of the fluids. Copyright © 2007 John Wiley & Sons, Ltd.     

基于无网格Galerkin方法的整体式柔性机构的多准则拓扑优化设计

基于一种新型的数值计算技术—无网格伽辽金法,提出了一种整体式柔性机构拓扑优化设计的新方法.利用移动最小二乘形函数和伽辽金的弱变分形式建立弹性问题的控制方程,用Lagrange乘子法增强本质边界条件.在优化问题中,同时综合考虑机构的柔性和结构的刚度要求,用折衷规划法建立了柔性机构拓扑优化的多准则优化模型,这样对于非凸的优化问题也能保证搜索到Pareto解集所有的有效解.基于SIMP密度函数惩罚模型和优化准则法,建立了一种设计变量的显式迭代格式.运用经典算例证明了文中方法的正确性和有效性.     

Chaos control of a nonlinear oscillator with shape memory alloy using an optimal linear control: Part I: Ideal energy source

In this paper, an optimal linear control is applied to control a chaotic oscillator with shape memory alloy (SMA). Asymptotic stability of the closed-loop nonlinear system is guaranteed by means of a Lyapunov function, which can clearly be seen to be the solution of the Hamilton–Jacobi–Bellman equation, thus guaranteeing both stability and optimality. This work is presented in two parts. Part I considers the so-called ideal problem. In the ideal problem, the excitation source is assumed to be an ideal harmonic excitation.     

Simulation of two‐phase flow–body interaction problems using direct forcing/fictitious domain–level set method

In the present paper, a direct forcing/fictitious domain (DF/FD)–level set method is proposed to simulate the twophase flow–body interaction. The DF/FD does not sacrifice accuracy and robustness by employing a discrete δ (Dirac delta) function to transfer quantities between the Eulerian nodes and Lagrangian points explicitly as the immersed boundary method. The advantages of this approach are the simple concept, the easy implementation and the utilization of original governing equation without modification. The main idea is to combine DF/FD method with the level set method in the Cartesian coordinates. We present the results of a number of test cases to illustrate the effectiveness of the proposed method for single‐phase flow–body interaction problem and the two‐phase flows with a stationary body. Eventually, the simulations of various water entry problems have been conducted to validate the capability and the accuracy of the present method on solving the twophase flow–body interaction. Consequently, the present results are found to be in good agreement with those of previous studies. Copyright © 2013 John Wiley & Sons, Ltd.     

Double Obstacle Problems with Obstacles Given by Non-C 2 Hamilton–Jacobi Equations

We prove optimal regularity for double obstacle problems when obstacles are given by solutions to Hamilton–Jacobi equations that are not C ². When the Hamilton–Jacobi equation is not C ² then the standard Bernstein technique fails and we lose the usual semi-concavity estimates. Using a non-homogeneous scaling (different speeds in different directions) we develop a new pointwise regularity theory for Hamilton–Jacobi equations at points where the solution touches the obstacle. A consequence of our result is that C ¹-solutions to the Hamilton–Jacobi equation $$\pm |\nabla h-a(x)|^2=\pm 1\,{\rm in}\,B_1,\quad h=f \,{\rm on}\, \partial B_1$$ , are, in fact, C ^1,α/2, provided that ${a \in C^\alpha}$ . This result is optimal and, to the authors’ best knowledge, new.     

An assessment of a parallel,finite element method for three‐dimensional,moving‐boundary flows driven by capillarity for simulation of viscous sintering

A parallel, finite element method is presented for the computation of three‐dimensional, free‐surface flows where surface tension effects are significant. The method employs an unstructured tetrahedral mesh, a front‐tracking arbitrary Lagrangian–Eulerian formulation, and fully implicit time integration. Interior mesh motion is accomplished via pseudo‐solid mesh deformation. Surface tension effects are incorporated directly into the momentum equation boundary conditions using surface identities that circumvent the need to compute second derivatives of the surface shape, resulting in a robust representation of capillary phenomena. Sample results are shown for the viscous sintering of glassy ceramic particles. The most serious performance issue is error arising from mesh distortion when boundary motion is significant. This effect can be severe enough to stop the calculations; some simple strategies for improving performance are tested. Copyright © 2001 John Wiley & Sons, Ltd.     

Optimum shape and topology design using the boundary element method

A procedure is developed for simultaneous shape and topology design optimization of linear elastic two-dimensional continuum structures. An intuitive approach is presented to treat such topological problems whereby material is eliminated from within the structure (by introducing holes at regions of low stress) through a sequence of shape optimization processes. A mathematical programming technique coupled with the boundary element (BE) method of response and sensitivity analyses enables the optimal positioning of these holes plus optimization of the overall structural shape. The analytical derivative BE formulation is explained together with the use of appropriate design velocity fields, and example problems are solved to demonstrate the optimization procedure.     

A semi‐Lagrangian level set method for incompressible Navier–Stokes equations with free surface

In this paper, we formulate a level set method in the framework of finite elements‐semi‐Lagrangian methods to compute the solution of the incompressible Navier–Stokes equations with free surface. In our formulation, we use a quasi‐monotone semi‐Lagrangian scheme, which is both unconditionally stable and essentially non oscillatory, to compute the advective terms in the Navier–Stokes equations, the transport equation and the equation of the reinitialization stage for the level set function. The method we propose is quite robust and flexible with regard to the mesh and the geometry of the domain, as well as the magnitude of the Reynolds number. We illustrate the performance of the method in several examples, which range from a benchmark problem to test the volume conservation property of the method to the flow past a NACA0012 foil at high Reynolds number. Copyright © 2005 John Wiley & Sons, Ltd.     

Nonlinear buckling optimization of composite structures considering “worst” shape imperfections

Nonlinear buckling optimization is introduced as a method for doing laminate optimization on generalized composite shell structures exhibiting nonlinear behaviour where the objective is to maximize the buckling load. The method is based on geometrically nonlinear analyses and uses gradient information of the nonlinear buckling load in combination with mathematical programming to solve the problem. Thin-walled optimal laminated structures may have risk of a relatively high sensitivity to geometric imperfections. This is investigated by the concepts of “worst” imperfections and an optimization method to determine the “worst” shape imperfections is presented where the objective is to minimize the buckling load subject to imperfection amplitude constraints. The ability of the nonlinear buckling optimization formulation to solve the laminate problem and determine the “worst” shape imperfections is illustrated by several numerical examples of composite laminated structures and the application of both formulations gives useful insight into the interaction between laminate design and geometric imperfections.