Designing plates for minimum internal resonances

Internal resonance is a nonlinear phenomenon for a structure when the eigenfrequencies of the structure are commensurable or close to being commensurable. Using optimization we have the possibility to control the eigenfrequencies, i.e., move an eigenfrequency, maximize a given eigenfrequency, or maximize the gap between eigenfrequencies. It is therefore also possible to design a structure that is as free as possible of internal resonance up to mode of order n.We consider plates made of two materials. The designs depend on the boundary conditions and on the frequency range within which the plate should be as free of internal resonance as possible. The two materials can either be two physical materials, or one can be a physical material and the other a weakening of the first material. By doing this we are in principle solving three different problems: a reinforcement problem, a problem of where to put holes in the structure, and, finally the more involved case (from a manufacturing point of view), of two different materials. The optimizations are performed using the finite element method for analysis and the topology optimization approach for design. The optimization problem is formulated using a bound formulation where the objective is to maximize a minimum detuning parameter. Special attention is given to the formulation of the conditions for internal resonance. Using the method presented in this paper it is possible to remove an unwanted nonlinear phenomenon without the use of a nonlinear model and without knowledge of the nonlinearities present in the system.     

A computational paradigm for multiresolution topology optimization (MTOP)

This paper presents a multiresolution topology optimization (MTOP) scheme to obtain high resolution designs with relatively low computational cost. We employ three distinct discretization levels for the topology optimization procedure: the displacement mesh (or finite element mesh) to perform the analysis, the design variable mesh to perform the optimization, and the density mesh (or density element mesh) to represent material distribution and compute the stiffness matrices. We employ a coarser discretization for finite elements and finer discretization for both density elements and design variables. A projection scheme is employed to compute the element densities from design variables and control the length scale of the material density. We demonstrate via various two- and three-dimensional numerical examples that the resolution of the design can be significantly improved without refining the finite element mesh.     

基于无网格局部Petrov-Galekin法的板结构拓扑优化

为将无网格法的优势集成到结构拓扑优化中,基于无网格局部Petrov-Galerkin(Meshless Local Petrov-Galerkin,MLPG)法进行板结构的拓扑优化.基于带惩罚的各向同性固体微结构(Solid Isotropic Microstructure with Penalization,SIMP...     

基于修正变分法开孔板和加强板结构的自由振动分析

为研究船舶开孔板和加强板结构的振动特性,用1阶剪切变形板理论描述各向同性板的位移场,并采用修正变分原理和区域分解方法建立板的离散动力学模型.每一块子域板的位移和转角分量通过第一类切比雪夫正交多项式展开.针对加强板模型,将该方法获得的结果与已经发表的文献和有限元商用软件计算结果进行对比,验证该方法的收敛性和正确性.基于修正变分法探讨多种开孔和加强板模型的自由振动特性,充分说明该数理模型和半解析方法是一种适合处理复杂板结构问题的数值工具.     

Robust topology optimization accounting for misplacement of material

The use of topology optimization for structural design often leads to slender structures. Slender structures are sensitive to geometric imperfections such as the misplacement or misalignment of material. The present paper therefore proposes a robust approach to topology optimization taking into account this type of geometric imperfections. A density filter based approach is followed, and translations of material are obtained by adding a small perturbation to the center of the filter kernel. The spatial variation of the geometric imperfections is modeled by means of a vector valued random field. The random field is conditioned in order to incorporate supports in the design where no misplacement of material occurs. In the robust optimization problem, the objective function is defined as a weighted sum of the mean value and the standard deviation of the performance of the structure under uncertainty. A sampling method is used to estimate these statistics during the optimization process. The proposed method is successfully applied to three example problems: the minimum compliance design of a slender column-like structure and a cantilever beam and a compliant mechanism design. An extensive Monte Carlo simulation is used to show that the obtained topologies are more robust with respect to geometric imperfections.     

A performance index for topology and shape optimization of plate bending problems with displacement constraints

This paper presents a performance index for topology and shape optimization of plate bending problems with displacement constraints. The performance index is developed based on the scaling design approach. This performance index is used in the Performance-Based Optimization (PBO) method for plates in bending to keep track of the performance history when inefficient material is gradually removed from the design and to identify optimal topologies and shapes from the optimization process. Several examples are provided to illustrate the validity and effectiveness of the proposed performance index for topology and shape optimization of bending plates with single and multiple displacement constraints under various loading conditions. The topology optimization and shape optimization are undertaken for the same plate in bending, and the results are evaluated by using the performance index. The proposed performance index is also employed to compare the efficiency of topologies and shapes produced by different optimization methods. It is demonstrated that the performance index developed is an effective indicator of material efficiency for bending plates. From the manufacturing and efficient point of view, the shape optimization technique is recommended for the optimization of plates in bending. Received November 27, 1998?Revised version received June 6, 1999     

Spatial orientation and topology optimization of modular trusses

Modularity in structural engineering offers significant cost advantages since identical components can be mass-produced in high quality-controlled facilities. While prior research has investigated the optimization of modular structures, this work is limited to the module topology periodicity, leading to solutions where some structure parts remain inefficiently used. This paper proposes a continuum-based formulation for optimizing simultaneously the module topology and spatial orientation in modular trusses. The numerical difficulties associated with the module rotations in the optimization formulation are identified, and properly handled by a novel topology-based rotation representation using group theory. A relaxation strategy based on complementary constraints enables the continuous integration of the module rotations in a lower-bound plastic design formulation (or fully stressed design). Case studies involving high density ground structures demonstrate that it is able in the worst case to retrieve the results obtained by imposing the module orientation a priori, and to improve the optimized design when freedom on the rotations is considered.     

Topology optimization using the p-version of the finite element method

A multiresolution topology optimization approach is proposed using the p-version finite element method (p-version FEM). Traditional topology optimization, where a density design variable is assigned to each element, is suitable for low-order h-version FEM. However, it cannot take advantage of the higher accuracy of higher-order p-version FEM analysis for generating results with higher resolution. In contrast, the proposed method separates density variables and finite elements so that the resolution of the density field, which defines the structure, can be higher than that of the finite element mesh. Thus, the method can take full advantage of the higher accuracy of p-version FEM.     

Topology optimization of three-dimensional structures using optimum microstructures

In this paper, optimum three-dimensional microstructures derived in explicit analytical form by Gibianski and Cherkaev (1987) are used for topology optimization of linearly elastic three-dimensional continuum structures subjected to a single case of static loading. For prescribed loading and boundary conditions, and subject to a specified amount of structural material within a given three-dimensional design domain, the optimum structural topology is determined from the condition of maximum integral stiffness, which is equivalent to minimum elastic complicance or minimum total elastic energy at equilibrium.The use of optimum microstructures in the present work renders the local topology optimization problem convex, and the fact that local optima are avoided implies that we can develop and present a simple sensitivity based numerical method of mathematical programming for solution of the complete optimization problem.Several examples of optimum topology designs of three-dimensional structures are presented at the end of the paper. These examples include some illustrative full three-dimensional layout and topology optimization problems for plate-like structures. The solutions to these problems are compared to results obtained earlier in the literature by application of usual two-dimensional plate theories, and clearly illustrate the advantage of the full three-dimensional approach.     

On the (non-)optimality of Michell structures

Optimal analytical Michell frame structures have been extensively used as benchmark examples in topology optimization, including truss, frame, homogenization, density and level-set based approaches. However, as we will point out, partly the interpretation of Michell’s structural continua as discrete frame structures is not accurate and partly, it turns out that limiting structural topology to frame-like structures is a rather severe design restriction and results in structures that are quite far from being stiffness optimal. The paper discusses the interpretation of Michell’s theory in the context of numerical topology optimization and compares various topology optimization results obtained with the frame restriction to cases with no design restrictions. For all examples considered, the true stiffness optimal structures are composed of sheets (2D) or closed-walled shell structures (3D) with variable thickness. For optimization problems with one load case, numerical results in two and three dimensions indicate that stiffness can be increased by up to 80 % when dropping the frame restriction. For simple loading situations, studies based on optimal microstructures reveal theoretical gains of +200 %. It is also demonstrated how too coarse design discretizations in 3D can result in unintended restrictions on the design freedom and achievable compliance.     

Evolutionary natural frequency optimization of thin plate bending vibration problems

This paper extends the evolutionary structural optimization method to the solution for maximizing the natural frequencies of bending vibration thin plates. Two kinds of constraint conditions are considered in the evolutionary structural optimization method. If the weight of a target structure is set as a constraint condition during the natural frequency optimization, the optimal structural topology can be found by removing the most ineffectively used material gradually from the initial design domain of a structure until the weight requirement is met for the target structure. However, if the specific value of a particular natural frequency is set as a constraint condition for a target structure, the optimal structural topology can be found by using a design chart. This design chart describes the evolutionary process of the structure and can be generated by the information associated with removing the most inefficiently used material gradually from the initial design domain of a structure until the minimum weight is met for maintaining the integrity of a structure. The main advantage in using the evolutionary structural optimization method lies in the fact that it is simple in concept and easy to be included into existing finite element codes. Through applying the extended evolutionary structural optimization method to the solution for the natural frequency optimization of a thin plate bending vibration problem, it has been demonstrated that the extended evolutionary structural optimization method is very useful in dealing with structural topology optimization problems.     

Design of conjugate, conjoined shapes and tilings using topology optimization

We present a method for obtaining conjugate, conjoined shapes and tilings in the context of the design of structures using topology optimization. Optimal material distribution is achieved in topology optimization by setting up a selection field in the design domain to determine the presence/absence of material there. We generalize this approach in this paper by presenting a paradigm in which the material left out by the selection field is also utilised. We obtain conjugate shapes when the region chosen and the region left-out are solutions for two problems, each with a different functionality. On the other hand, if the left-out region is connected to the selected region in some pre-determined fashion for achieving a single functionality, then we get conjoined shapes. The utilization of the left-out material, gives the notion of material economy in both cases. Thus, material wastage is avoided in the practical realization of these designs using many manufacturing techniques. This is in contrast to the wastage of left-out material during manufacture of traditional topology-optimized designs. We illustrate such shapes in the case of stiff structures and compliant mechanisms. When such designs are suitably made on domains of the unit cell of a tiling, this leads to the formation of new tilings which are functionally useful. Such shapes are not only useful for their functionality and economy of material and manufacturing, but also for their aesthetic value.     

Structural topology optimization based on non-local Shepard interpolation of density field

This paper presents a non-local density interpolation strategy for topology optimization based on nodal design variables. In this method, design variable points can be positioned at any locations in the design domain and may not necessarily coincide with elemental nodes. By using the Shepard family of interpolants, the density value of any given computational point is interpolated by design variable values within a certain circular influence domain of the point. The employed interpolation scheme has an explicit form and satisfies non-negative and range-restricted properties required by a physically significant density interpolation. Since the discretizations of the density field and the displacement field are implemented on two independent sets of points, the method is well suited for a topology optimization problem with a design domain containing higher-order elements or non-quadrilateral elements. Moreover, it has the ability to yield mesh-independent solutions if the radius of the influence domain is reasonably specified. Numerical examples demonstrate the validity of the proposed formulation and numerical techniques. It is also confirmed that the method can successfully avoid checkerboard patterns as well as “islanding” phenomenon.     

Topological sensitivity derivative and finite topology modifications: application to optimization of plates in bending

The concept of topological sensitivity derivative is introduced and applied to study the problem of optimal design of structures. It is assumed, that virtual topology variation is described by topological parameters. The topological derivative provides the gradients of objective functional and constraints with respect to these parameters. This derivative enables formulation of the conditions of topology transformation. In this paper formulas for the topological sensitivity derivative for bending plates are derived. Next, the topological derivative is used in the optimization process in order to formulate conditions of finite topology modifications and in order to localize positions of the modifications. In the case of plates they are related to introduction of holes and introduction of stiffeners. The theoretical considerations are illustrated by some numerical examples.     

Optimal topology design of structures under dynamic loads

When elastic structures are subjected to dynamic loads, a propagation problem is considered to predict structural transient response. To achieve better dynamic performance, it is important to establish an optimum structural design method. Previous work focused on minimizing the structural weight subject to dynamic constraints on displacement, stress, frequency, and member size. Even though these methods made it possible to obtain the optimal size and shape of a structure, it is necessary to obtain an optimal topology for a truly optimal design. In this paper, the homogenization design method is utilized to generate the optimal topology for structures and an explicit direct integration scheme is employed to solve the linear transient problems. The optimization problem is formulated to find the best configuration of structures that minimizes the dynamic compliance within a specified time interval. Examples demonstrate that the homogenization design method can be extended to the optimal topology design method of structures under impact loads.Presented at WCSMO-2, held in Zakopane, Poland, 1997     

Geometry and topology optimization of plane frames for compliance minimization using force density method for geometry model

A new method is proposed for simultaneous optimization of shape, topology and cross section of plane frames. Compliance against specified loads is minimized under constraint on structural volume. Difficulties caused by the melting nodes can be alleviated to some extent by introducing force density as design variables for defining the geometry, where the side constraints are assigned for force density to indirectly avoid the existence of extremely short members. Force density method is applied to an auxiliary cable-net model with different boundary and loading conditions so that the regularity of force density matrix is ensured by positive force densities. Sensitivity coefficients of the objective and constraint functions with respect to the design variables are also explicitly calculated. After the optimal geometry of the frame is obtained, the topology is further improved by removing the thin members and combining closely spaced nodes. It is demonstrated in the numerical examples of three types of frames that rational geometry and topology can be achieved using the proposed method, and the effect of bending moment on the optimal solution is also discussed.

     

Two-dimensional time-dependent vortex regions based on the acceleration magnitude

Acceleration is a fundamental quantity of flow fields that captures Galilean invariant properties of particle motion. Considering the magnitude of this field, minima represent characteristic structures of the flow that can be classified as saddle- or vortex-like. We made the interesting observation that vortex-like minima are enclosed by particularly pronounced ridges. This makes it possible to define boundaries of vortex regions in a parameter-free way. Utilizing scalar field topology, a robust algorithm can be designed to extract such boundaries. They can be arbitrarily shaped. An efficient tracking algorithm allows us to display the temporal evolution of vortices. Various vortex models are used to evaluate the method. We apply our method to two-dimensional model systems from computational fluid dynamics and compare the results to those arising from existing definitions.     

Optimization methods for truss geometry and topology design

Truss topology design for minimum external work (compliance) can be expressed in a number of equivalent potential or complementary energy problem formulations in terms of member forces, displacements and bar areas. Using duality principles and non-smooth analysis we show how displacements only as well as stresses only formulations can be obtained and discuss the implications these formulations have for the construction and implementation of efficient algorithms for large-scale truss topology design. The analysis covers min-max and weighted average multiple load designs with external as well as self-weight loads and extends to the topology design of reinforcement and the topology design of variable thickness sheets and sandwich plates. On the basis of topology design as an inner problem in a hierarchical procedure, the combined geometry and topology design of truss structures is also considered. Numerical results and illustrative examples are presented.