A shape optimisation method of a body located in adiabatic flows

The purpose of this study is to derive an optimal shape of a body located in adiabatic flow. In this study, we use the equation of motion, the equation of continuity and the pressure–density relation derived from the Poisson’s law as the governing equation. The formulation is based on an optimal control theory in which a performance function of fluid force is taken into consideration. The performance function should be minimised satisfying the governing equations. This problem can be solved without constraints by using the adjoint equation with adjoint variables corresponding to the state equation. The performance function is defined by the drag and lift forces acting on the body. The weighted gradient method is applied as a minimisation technique, the Galerkin finite element method is used as a spatial discretisation and the implicit scheme is used as a temporal discretisation to solve the state equations. The mixed interpolation, the bubble function for velocity and the linear function for density, is employed as the interpolation. The optimal shape is obtained for a body in adiabatic flows.     

Shape optimisation of an oscillating body in fluid flow by adjoint equation and ALE finite element methods

The purpose of this paper is to determine the shape of an oscillating body by minimising drag and lift forces, located in a transient incompressible viscous fluid flow by means of the Arbitrary Lagrangian Eulerian finite element method and an optimal control theory. A performance function is expressed by the drag and lift forces. The performance function should be minimised satisfying the state equation and the constant volume condition. Therefore, this problem can be transformed into a minimisation problem without constraint by the Lagrange multiplier method. The adjoint equation and the gradient of the performance function are used to update the shape of the body. In this study, as a minimisation technique, the weighted gradient method is applied. The final shape is obtained of which drag and lift forces are reduced by 66.2% and 92.8%, respectively. The final shape obtained by this study is compared with the final shape of the non-oscillating body. The obtained final shape of the oscillating body is significantly different from the non-oscillating body.     

Simulation of liquid sloshing in curved‐wall containers with arbitrary Lagrangian–Eulerian method

There are many challenges in the numerical simulation of liquid sloshing in horizontal cylinders and spherical containers using the finite element method of arbitrary Lagrangian–Eulerian (ALE) formulation: tracking the motion of the free surface with the contact points, defining the mesh velocity on the curved wall boundary and updating the computational mesh. In order to keep the contact points slipping along the curved side wall, the shape vector in each time advancement is defined to modify the kinematical boundary conditions on the free surface. A special function is introduced to automatically smooth the nodal velocities on the curved wall boundary based on the liquid nodal velocities. The elliptic partial differential equation with Dirichlet boundary conditions can directly rezone the inner nodal velocities in more than a single freedom. The incremental fractional step method is introduced to solve the finite element liquid equations. The numerical results that stemmed from the algorithm show good agreement with experimental phenomena, which demonstrates that the ALE method provides an efficient computing scheme in moving curved wall boundaries. This method can be extended to 3D cases by improving the technique to compute the shape vector. Copyright © 2007 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.     

Optimal control of shallow water flow using time-delay function

This paper presents an optimal control system that includes a time-delay function for application to flood control setups with a retardation area. This system consists of the present and past controls that express flow behaviour in the retardation area. Optimal control theory is used to obtain a control discharge that satisfies the state equation including the time-delay function and minimizes the performance function. The optimal control and the delayed control discharges are obtained by the solution of an adjoint equation. The weighted gradient method is employed as a minimization algorithm. The Galerkin finite element procedure is employed to discretize the state and adjoint equations in the spatial direction. The bubble function interpolation, originated by the authors' group, using a stabilized term, is employed for the discretization in space. The flood flow in the Tsurumi river is presented as a numerical model. We show in this paper that floods can be controlled by means of a time-delay function.     

基于虚荷载变量的形状优化和灵敏度分析

基于选择施加在结构“控制点”上的虚荷载作为优化设计变量,针对一种新的承受约束的形状优化数值方法进行了研究。借助于节点位移与虚荷载之间的线性关系,提出了一种新的计算灵敏度系数的解析方法。利用节点移动速度域概念构造了优化新形状产生的计算公式,以结构中节点的最大应力最小化作为优化目标,通过控制网格结点的最大位移量,较好地解决了单元网格在形状优化中的扭曲问题。对三个不同的实例成功地完成了形状优化。     

On the formulation of the planar ANCF triangular finite elements

In this paper, new planar isoparametric triangular finite elements (FE) based on the absolute nodal coordinate formulation (ANCF) are developed. The proposed ANCF elements have six coordinates per node: two position coordinates that define the absolute position vector of the node and four gradient coordinates that define vectors tangent to coordinate lines (parameters) at the same node. To shed light on the importance of the element geometry and to facilitate the development of some of the new elements presented in this paper, two different parametric definitions of the gradient vectors are used. The first parametrization, called area parameterization, is based on coordinate lines along the sides of the element in the reference configuration, while the second parameterization, called Cartesian parameterization, employs coordinate lines defined along the axes of the structure (body) coordinate system. The fundamental differences between the ANCF parameterizations used in this investigation and the parametrizations used for conventional finite elements are highlighted. The Cartesian parameterization serves as a unique standard for the triangular FE assembly. To this end, a transformation matrix that defines the relationship between the area and the Cartesian parameterizations is introduced for each element in order to allow for the use of standard FE assembly procedure and define the structure (body) inertia and elastic forces. Using Bezier geometry and a linear mapping, cubic displacement fields of the new ANCF triangular elements are systematically developed. Specifically, two new ANCF triangular finite elements are developed in this investigation, namely four-node mixed-coordinate and three-node ANCF triangles. The performance of the proposed new ANCF elements is evaluated by comparison with the conventional linear and quadratic triangular elements as well as previously developed ANCF rectangular and triangular elements. The results obtained in this investigation show that in the case of small and large deformations as well as finite rotations, all the elements considered can produce correct results, which are in a good agreement if appropriate mesh sizes are used.     

Shape optimisation of wall structures located in solitary wave flows

ABSTRACT

In this paper, we present a shape optimisation method for wall structures due to the wave force induced by a solitary wave. The fluid is assumed to be incompressible. Introducing the adiabatic assumption in addition, the acoustic velocity method presented by the author's group, the SUPG finite element method, is effectively used. To evaluate the wave force, we use the performance functional, which consists of the sum of the square of the wave force integrated between the starting and final times. The coordinates of the wall structure are regulated to obtain the minimum performance functional. The adjoint equation method is utilised to derive the gradient of the performance functional with respect to the coordinates. The simple weighted gradient method is employed as the minimisation procedure. Two numerical studies show that the results are consistent with existing structures and provide useful information on the practical design of coastal structures.     

Fractal patterns from the dynamics of combined polynomial root finding methods

A solid tetrahedral finite element employing the absolute nodal coordinate formulation (ANCF) is presented. In the ANCF, the mass matrix and vector of the generalized gravity forces used in the equations of motion are constant, whereas the vector of the elastic forces is highly nonlinear. The proposed solid element uses translations of nodes as sets of nodal coordinates. The tetrahedral shape of the element makes it suitable for modeling structures with complex shapes, and the small number of the degrees of freedom enables good performance and versatile application to problems of structural dynamics. The accuracy and convergence of the element were investigated using statics and dynamics benchmarks and a practical industry application.     

局部有限元模型边界条件的直接传递方法

提出一种将整体分析得到的节点力或节点位移直接传递到精细化局部有限元模型的方法,即部分混合单元法。沿精细化局部有限元模型周边建立一组过渡单元,该组过渡单元采用与整体模型一致的单元类型和模拟方式,其外侧边界上的节点与整体模型节点的相对坐标对应,内侧边界与精细化局部有限元模型采用基于面约束的方式连接。在外侧边界上根据节点坐标对应施加整体分析获得的节点力或节点位移,过渡单元就可直接将边界条件传递到精细化局部有限元模型。通过贵州红水河特大桥钢-混结合段的精细化有限元分析,验证了本文方法的实用性和有效性。     

Numerical solution of steady free‐surface flows by the adjoint optimal shape design method

Numerical solution of flows that are partially bounded by a freely moving boundary is of great importance in practical applications such as ship hydrodynamics. Free‐boundary problems can be reformulated into optimal shape design problems, which can in principle be solved efficiently by the adjoint method. In this work we investigate the suitability of the adjoint shape optimization method for solving steady free‐surface flows. The asymptotic convergence behaviour of the method is determined for free‐surface flows in 2D and 3D. It is shown that the convergence behaviour depends sensitively on the occurrence of critical modes. The convergence behaviour is moreover shown to be mesh‐width independent, provided that proper preconditioning is applied. Numerical results are presented for 2D flow over an obstacle in a channel. The observed convergence behaviour is indeed mesh‐width independent and conform the derived asymptotic estimates. Copyright © 2003 John Wiley & Sons, Ltd.     

基于遗传算法和梯度算法压电层合结构的最优形状控制

基于压电层合结构的有限元方程,运用ANSYS/APDL语言,编制了力-电耦合有限元分析程序(MPFEMP).以该程序为计算基础,采用遗传算法和一阶梯度优化算法,以压电片尺寸为设计变量,以压电层合梁和板的预期位移或最小重量为目标函数,给定初始变量和适应度函数,通过循环迭代MPFEMP计算程序,研究了多点控制的压电层合梁板结构的形状最优控制.结论对比分析证明了两种优化方法分析压电层合结构的有效性,同时,对复杂多层智能结构的最优形状控制和主动控制研究具有一定的参考价值.     

Shape optimization of a body located in low Reynolds number flow

The purpose of this study is to perform a numerical application of the shape optimization formulation of a body located in an incompressible viscous flow field. The formulation is based on an optimal control theory in which a performance function of the fluid force is introduced. The performance function should be minimized satisfying the state equation. This problem can be transformed into the minimization problem without constraint condition by the Lagrange multiplier method and the adjoint equations using adjoint variables corresponding to the state equations. As a numerical study, the drag force minimization problem in the steady Stokes flow, which means approximated equation of the low Reynolds number Navier–Stokes equation is carried out. After that, the unsteady Navier–Stokes flow is analysed. As the minimization algorithm, the steepest descent method is successfully applied. Copyright © 2005 John Wiley & Sons, Ltd.     

Integration of B-spline geometry and ANCF finite element analysis

The goal of this investigation is to introduce a new computer procedure for the integration of B-spline geometry and the absolute nodal coordinate formulation (ANCF) finite element analysis. The procedure is based on developing a linear transformation that can be used to transform systematically the B-spline representation to an ANCF finite element mesh preserving the same geometry and the same degree of continuity. Such a linear transformation that relates the B-spline control points and the finite element position and gradient coordinates will facilitate the integration of computer aided design and analysis (ICADA). While ANCF finite elements automatically ensure the continuity of the position and gradient vectors at the nodal points, the B-spline representation allows for imposing a higher degree of continuity by decreasing the knot multiplicity. As shown in this investigation, a higher degree of continuity can be systematically achieved using ANCF finite elements by imposing linear algebraic constraint equations that can be used to eliminate nodal variables. The analysis presented in this study shows that continuity of the curvature vector and its derivative which corresponds in the cubic B-spline representation to zero knot multiplicity can be systematically achieved using ANCF finite elements. In this special case, as the knot multiplicity reduces to zero, the recurrence B-spline formula causes two segments to automatically blend together forming one cubic segment defined on a larger domain. Similarly in this special case, the algebraic constraint equations required for the C ³ continuity convert two ANCF cubic finite elements to one finite element, demonstrating the strong relationship between the B-spline representation and the ANCF finite element representation. For the same order of interpolation, higher degree of continuity at the finite element interface can lead to a coarser mesh and to a lower dimensional model. Using the B-spline/ANCF finite element transformation developed in this paper, the equations of motion of a finite element mesh that represents exactly the B-spline geometry can be developed. Because of the linearity of the transformation developed in this investigation, all the ANCF finite element desirable features are preserved; including the constant mass matrix that can be used to develop an optimum sparse matrix structure of the nonlinear multibody system dynamic equations.     

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.     

Shape Optimization of Body Located in Incompressible Viscous Flow Based on Optimal Control Theory

This paper presents a numerical method of shape optimization of a body located in an incompressible viscous flow described by the Stokes and Oseen equations. The purpose of this study is to find the optimal shape that minimizes the fluid forces subjected to the body. The formulation of the shape optimization is based on the optimal control theory. The first thing that should be carried out in the optimal control theory is to define a performance function, which expresses the optimal shape. In this study, the fluid forces minimization problem is treated, i.e. fluid forces are directly used in the performance function. The performance function must be minimized subject to the basic equation. The optimal shape, which minimizes the fluid force, is pursued in this paper. This problem can be transformed into the minimization problem without constraint conditions by the Lagrange multiplier. As a numerical example, drag force minimization problems of a body located in low Reynolds number flows are carried out.     

Water quality control by bank placement based on optimal control and finite element method

This paper presents a method for quality control by bank placement based on an optimal control theory and the finite element method. The shallow water equation is employed for the analysis of the flow condition and the advection‐diffusion equation is used for the analysis of pollutant concentration. The optimal control theory is utilized to obtain a control value for the objective state value. The shear‐slip mesh update method which is suitable for the rotational problem of body is employed. To solve the optimization problem, the time domain decomposition method is applied as a technique of storage requirements reduction. The Sakawa–Shindo method is employed as a minimization technique. The Crank–Nicolson method is applied to the temporal discretization. A method for optimal control of bank placement has been presented. Copyright © 2003 John Wiley & Sons, Ltd.     

节点梯度光滑有限元配点法

配点法构造简单、计算高效,但需要用到数值离散形函数的高阶梯度,而传统有限元形函数的梯度在单元边界处通常仅具有C0连续性,因此无法直接用于配点法分析.本文通过引入有限元形函数的光滑梯度,提出了节点梯度光滑有限元配点法.首先基于广义梯度光滑方法,定义了有限元形函数在节点处的一阶光滑梯度值,然后以有限元形函数为核函数构造了有...     

基于物理的变形曲线在结构形状优化中的应用

用各向异性模型定义NURBS曲线的变形能,基于所提出的变形能模型,用有限元法对NURBS曲线表达的设计边界进行模态计算,然后,将设计边界用模态向量的线性组合参数化表示。将这种基于边界特征向量的几何形状表示方法应用于优化参数定义．提出了一种适用于结构形状优化的自适应几何精化方法,它有效地将户型有限元分析、优化方法和设计边界的形状表达集成在一起。