Truss-like Continuum Research Articles

Non-intrusive polynomial chaos expansion for robust topology optimization of truss-like continuum under random loads

Non-intrusive polynomial chaos expansion for robust topology optimization of truss-like continuum under random loads

Robust topology optimization of truss-like continuum structures under uncertain loads based on cloud model

The traditional structural performance may be seriously degraded due to applied indeterministic cases. A novel framework for robust topology optimization of truss-like continuum considering uncertain loads is presented. The robust optimization formula is constructed as minimizing the weighted sum of expectancy and standard variance of structural compliance with volume constraints. In this framework, both the magnitude and direction of applied loads are uncertain and independent. Truss-like continuum provides the topology optimal structure theoretically. Based on the classic truss-like optimization method, the statistical moments of compliance are accurately evaluated utilizing a bivariate cloud model tool. An effective strategy based on linear decoupling is reported to largely reduce the total number of finite element analysis. The optimization is achieved with gradient-based mathematic programming. Several numerical examples demonstrate a better structural robust performance obtained by the proposed algorithm than the traditional one.

Topology optimization of truss support systems for a point- supported glass curtain wall based on truss-like continua

Steel truss structures for point-supported glass curtain walls with a web member configuration are presented using topology optimization based on truss-like continua. The initial structure comprises chords and partial web members. The initial design domain, surrounded by the members of the initial structure, is filled with a truss-like continuum, which is used to simulate the distributive web members of a steel truss. The structure members, together with the truss-like continuum, are analysed and calculated under horizontal point forces. The densities and orientations of the truss-like members at every node are taken as design variables and optimized. The members in the truss-like continuum are optimized using the fully stressed criterion. The topology optimization problem of minimum compliance with volume constraints is solved. Thus, an optimal web member configuration that suitably reinforces the lateral stiffness of the structure is obtained.

A mathematical programming method for the topology optimization of a truss-like continuum

A mathematical programming method for the topology optimization of a truss-like continuum

Seismic analysis of steel structure with brace configuration using topology optimization

Seismic analysis for steel frame structure with brace configuration using topology optimization based on truss-like material model is studied. The initial design domain for topology optimization is determined according to original steel frame structure and filled with truss-like members. Hence the initial truss-like continuum is established. The densities and orientation of truss-like members at any point are taken as design variables in finite element analysis. The topology optimization problem of least-weight truss-like continuum with stress constraints is solved. The orientations and densities of members in truss-like continuum are optimized and updated by fully-stressed criterion in every iteration. The optimized truss-like continuum is founded after finite element analysis is finished. The optimal bracing system is established based on optimized truss-like continuum without numerical instability. Seismic performance for steel frame structures is derived using dynamic time-history analysis. A numerical example shows the advantage for frame structures with brace configuration using topology optimization in seismic performance.

Topology optimization of bracing systems using a truss-like material model

To minimize the compliance of frame, a method to optimize the topology of bracing system in a frame is presented. The frame is first filled uniformly with a truss-like continuum, in which there are an infinite number of members. The frame and truss-like continuum are analysed by the finite element method altogether. By optimizing the distribution of members in the truss-like continuum over the whole design domain, the optimal bracing pattern is determined. As a result, the frame\'s lateral stiffness is enforced. Structural compliance and displacement are decreased greatly with a smaller increase in material volume. Since optimal bracing systems are described by the distribution field of members, rather than by elements, fewer elements are needed to establish the detailed structure. Furthermore, no numerical instability exists. Therefore it has high calculation effectiveness.

Topology Optimization of Frame Bracing System for Natural Frequency

A method to optimize the topology of frame bracing system is presented. Firstly frame is filled by truss-like continuum uniformly. The truss-like continuum simulates the distributive bracing system. The frame combined with truss-like continuum is analyzed by finite element method. The densities and orientations of bracing system at nodes are taken as design variables. The distribution field of bracing system is optimized by the method of moving asymptotes (MMA) and the method of steepest descent. The frame bracing system is established according to the optimal distribution field of bracing system. The natural frequency of optimal braced frame increases to more than two times while material increases only 5%. For no intermediate densities being suppressed, there is no numerical instability, such as checkerboard patterns and one-node connected hinges. The expression of natural frequency and its sensitivities of truss-like continuum are derived.

Topology optimization of truss-like continuum structures for natural frequencies

A method to maximize the natural frequencies of vibration of truss-like continua with the constraint of material volume is presented. Truss-like is a kind of particular anisotropic continuum, in which there are finite numbers of members with infinitesimal spaces. Structures are analyzed by finite element method. The densities and orientations of members at nodes are taken as design variables. The densities and orientations of members in elements are interpolated by these values at nodes; therefore they vary continuously in design domain. For no intermediate densities being suppressed, there is no numerical instability, such as checkerboard patterns and one-node connected hinges. The natural frequency and its sensitivities of truss-like continuum are derived. Optimization is achieved by the techniques of moving asymptotes and steepest descent. Several numerical examples are provided to demonstrate this optimization method.

Topology Optimization of Truss-Like Continuum under Stress Constraints by Sequence Linear Program

A method to minimize material volume of truss-like continuum with stress constraint under multi-loads cases by sequence linear program was presented. It is assumed that there were infinite numbers of members with infinitesimal spaces along two orthotropic orientations at any position. A truss-like material model was employed to simulate the topological optimal structures. Members were assumed as distributed continuously in design domain. Structure was analyzed by finite element method. The densities and orientations of members at nodes were taken as designed variables. Their values in elements vary continuously. An iterative procedure was suggested to optimize the distributed field of members. The sensitivities of stresses constraints were derived. Truss-like continuum was built by the method in this paper.

Optimal Grillages by 3-Nodes Triangle Finite Element

By triangle finite elements, grillages with stress constraints are optimized. From it an optimal beams systems or plate with reinforced ribs is obtained. A truss-like continuum material is adapted. The material distribution field in design domain is optimized by fully-stressed criterion. The densities and orientations of the beams or reinforced ribs at nodes in grillages are taken as design variables. These densities and orientations vary in design domain continuously. The optimal distribution fields of moments, deformation and material are obtained simultaneously. Subsequently the discrete structures are determined based on the optimal distribution fields. A very simple geometry matrix of triangle plate element is given. The stiffness matrix of truss-like continuum in triangle element is derived. The optimization iteration procedure is suggested. The optimization procedure is accomplished by computer program automatically.

Topology optimization of truss-like continua with three families of members model under stress constraints

A method to minimize structural volume under stress constraints subject to multiple load cases is presented. A new material model is developed and employed to simulate the constitutive relation of the truss-like continuum. It is assumed that there are infinite numbers of members with infinitesimal spaces along three orientations at any position. The densities and orientations of members at all nodes are taken as design variables. An iterative optimization method is presented. In one iteration step, design variables are optimized separately and independently. The orientations of three families of members at every node are optimized by mathematics program method. Fully stressed criterion is adapted to optimize member densities. In the every iteration step, member densities are adjusted to make their strains smaller than the permissible values while the stress states is assumed to keep unchanged. The densities and orientations of members at any point inside an element are obtained by their interpolating the corresponding values (i.e., the densities and orientations) at the nodes of the element. These values vary continuously inside an element and the intermediate values are not suppressed. By using this technique the optimal truss-like continuum is formed, which represents for member distribution field. Once parts of members are chosen, discrete truss can be constructed according to the continuous member distributive field. This discrete structure is a nearly optimal structure. In above process, there are no numerical instabilities such as checkerboard and mesh-dependency. Numerical examples are presented to demonstrate the effectiveness of the present method.

Topology optimization for minimum compliance under multiple loads based on continuous distribution of members

A method to minimize the compliance of structures subject to multiple load cases is presented. Firstly, the material distribution in design domain is optimized to form a truss-like continuum. The anisotropic composite is employed as the material model to simulate the constitutive relation of the truss-like continuum. The member densities and orientations at the nodes are taken as design variables. The member densities and orientations at any point in an element vary continuously. Then, parts of members, which are formed according to the member distribution field, are chosen to form the nearly optimum discrete structure. Lastly, the positions of the nodes and the cross-sectional areas of the members are optimized. In the above process, numerical instabilities such as checkerboard and mesh dependencies disappear without any additional technique. The sensitivities of the compliance are derived. Examples are presented to demonstrate the capability of the proposed method.

Forming Michell truss in three-dimensions by finite element method

The finite element method to form Michell truss in three-dimensions is presented. The orthotropic composite with fiber-reinforcement is employed as the material model to simulate Michell truss. The orientation and densities of fibers at nodes are taken as basic design variables. The stresses and strains at nodes are calculated by finite element method. An iteration scheme is suggested to adjust the orientations of fibers to be along the orientations of principal stresses, and the densities of fibers according to the strains in the orientations of fibers. The strain field satisfying Michell criteria and truss-like continuum are achieved after several iterations. Lastly, the Michell truss is showed by continuous lines, which are formed according to the orientations of fibers at nodes. Several examples are used to demonstrate the efficiency of the presented approach.

Optimal layout of cantilever trusses

For given allowable stress, Michell (Ref. 1) has investigated the optimal design of a cantilever truss that is to transmit a given load to two given fixed points of support. Disregarding the weight of the connections between the bars, he found that the truss of minimum weight is a truss-like continuum with an infinity of joints, and with bars that are mostly of infinitesimal length. In the present paper, a finite number of joints is enforced by including in the structural weight, which is to be minimized, not only the weight of the bars but also the weight of their connections, which is assumed to be proportional to the number of joints. The concept of two adjoint trusses is introduced, each of which coincides with the Maxwell diagram of the other truss. Two adjoint trusses have the same weight, and an optimal truss is therefore self-adjoint. The optimal configurations of 6-joint and 11-joint cantilever trusses are discussed, and the range of the weight of the typical joint is determined for which the 6-joint truss is optimal.