Continuum Topology Optimization Method Research Articles

Structural topology optimization of elastoplastic continuum under shakedown theory

AbstractThis article presents an elastoplastic continuum topology optimization method for shakedown under the SIMP‐based framework for the first time, aiming to maximize shakedown load‐carrying capacity under variable repeated loading. In contrast to most elastoplastic topology optimizations which have path‐dependent properties, the shakedown topology optimization only needs loading vertices of the loading domain rather than any complete loading history. Based on Melan's lower bound theorem, the gradient‐based topology optimization framework synthesizing the shakedown analysis and sensitivity analysis is developed to maximize the shakedown multiplier. Path‐independent sensitivity related to the specified density interpolation model is derived analytically for the first time in conjunction with the adjoint method. Besides, a primal‐dual algorithm of shakedown analysis is implemented for computing the shakedown load limit of structures and corresponding adjoint variables needed in the sensitivity analysis. In addition to the density interpolation scheme of material Young's module, the yield strength is also interpolated to overcome numerical difficulties. Several numerical examples demonstrate the effectiveness of the proposed method, which indicates the shakedown load‐carrying performance enhancements.

Stiffener layout optimization framework by isogeometric analysis-based stiffness spreading method

An innovative stiffener layout optimization framework is proposed based on the isogeometric analysis-based stiffness spreading method (IGA-based SSM), where the shape and size of stiffeners can be simultaneously optimized. The IGA-based SSM is employed to obtain the stiffness spreading matrices of the stiffeners that are embedded in the flat plate. The flat plate is modeled by high-order continuous isogeometric degenerated shell elements that can provide a smooth sensitivity field. Since the sensitivity required for the optimization can be obtained analytically, a gradient-based optimization algorithm can be employed. Moreover, the ground structure is used to provide the initial layouts of stiffeners and ensure the connectivity of stiffeners. A local geometry control strategy is also adopted to avoid the intersection of stiffeners in the optimization process. To verify the effectiveness and efficiency of the proposed method, typical illustrative examples are discussed in detail, considering the compliance minimization optimization problem. Compared with the continuum topology optimization method, the proposed method can provide a clear stiffener layout with a lower computational cost due to fewer design variables and fewer degrees of freedom of the optimization problem. Besides, the maximum thicknesses of stiffeners can be explicitly controlled, and it can easily solve the optimization problem of various volume fraction constraints. Finally, the proposed method is extended to solve a layout optimization problem of the engine nozzle adjusting plate under pressure loads, which obtains a good stiffener layout under the maximum displacement constraint.

Form-finding and shape optimization of bio-inspired branching structures based on graphic statics

Structure’s internal force transfer efficiency is the bases of pursuing the creative structural form. This paper reveals the mechanical mechanisms of the bio-inspired branching structures and further presents a numerical form-finding and shape optimization method for bio-inspired branching structures based on graphic statics with the constraints of strength, stiffness of boundary conditions are considered. Efficient structural form is obtained by connecting the internal force equilibrium and the structural performance in designing process. The reciprocal diagrams in graphic statics are used to solve the form-finding problem to generate compression or tension-only tree-like structure. Establishing the formula of the evaluation criteria of internal force transfer efficiency to characterize the structural performances, the criteria is compared with the continuum topology optimization method, the shape of tree-like structure is optimized by maximizing the structural efficiency under static loads. As a result, suitable external constraints can make a structure efficient and improve the structural redundancy, material properties determine the structural form, which in turn affects the structural behavior. Consequently, the interrelation mechanisms between material-property, structural-form, and structure-performance of the highly efficient bio-inspired branching structures are found. The internal force transfer efficiency and the strength and stiffness of boundary condition are the main factors to determine the optimal solution in the topological design exploration of load-bearing structure, these factors could be controlled to obtain the effective and elegant structural form for practical fabrication. The proposed numerical method could provide intuitive understanding and real-time feedback for the optimal solution, avoid the single solution obtained by the conventional optimization techniques and limited control to modify the output topology. Besides, more structural performance (e.g. buckling, vibration) can be considered in the structural designing process.

The robust fail-safe topological designs based on the von Mises stress

For large-scale equipment, e.g. aerospace and architecture industry, it is valuable to guarantee that one structure could survive partial damages. Due to the location of the damage is unknown in prior, results in a high number of failure scenarios to be calculated when considering fail-safe requirement in topology optimization. In this article, we propose an efficient continuum topology optimization method on the basis of the design philosophy of robust fail-safe structure. A number of patches with predefined shapes are used to simulate the material failure. The material properties of damaged models are interpolated by von Mises stress to construct the well-posed optimization model. The damaged compliance for the worst failure case is set as the optimization objective, and the KS function is adopted to approximate the non-differentiable max-operator. A computationally efficient sensitivity formulation is derived via the adjoint method. To suppress the highly nonlinear stress behavior and the phenomenon of optimization oscillation, an extended variable update scheme within the framework of Optimality Criteria (OC) method is developed. Representative benchmarks show that the presented strategy has the effectiveness at yielding the fail-safe structure by using acceptable computational cost.

An adaptive mesh‐adjustment strategy for continuum topology optimization to achieve manufacturable structural layout

SummaryThis paper presents an adaptive mesh adjustment algorithm for continuum topology optimization method to describe the structural boundary using nonuniform isoparametric element. A criterion on the basis of the node movement is proposed; herein, the densities and coordinates of the nodes are defined to instruct the deformation of finite elements in subsequent optimization iterations. With such a scheme, the topology optimization can start from a regular mesh discretization then gradually yields an optimal design with clear structural boundaries. The element in the transition along the boundary is refined; on the contrary, the pure solid or void element is coarsen. The contribution of this work is to improve the resolution of the structural boundaries and decrease the percentage of transitional regions with the invariant design variable. Several 2D and 3D numerical examples indicate the effectiveness of our proposed method. Seen from the examples, the structural boundary become smoother and the intermediate densities have been reduced up to 70%. In addition, a design process based on the presented method is proposed to make the optimum solutions be fabricated conveniently and accurately by linking it with the 3D design software, ie, SolidWorks, which is also demonstrated in the numerical examples.

Generating support structures for additive manufacturing with continuum topology optimization methods

PurposeThe purpose of this paper is to explore the possibility of combining additive manufacturing (AM) with topology optimization to generate support structures for addressing the challenging overhang problem. The overhang problem is considered as a constraint, and a novel algorithm based on continuum topology optimization is proposed.Design/methodology/approachA mathematical model is formulated, and the overhang constraint is embedded implicitly through a Heaviside function projection. The algorithm is based on the Solid Isotropic Material Penalization (SIMP) method, and the optimization problem is solved through sensitivity analysis.FindingsThe overhang problem of the support structures is fixed. The optimal topology of the support structures is developed from a mechanical perspective and remains stable as the material volume of support structures changes, which allows engineers to adjust the material volume to save cost and printing time and meanwhile ensure sufficient stiffness of the support structures. Three types of load conditions for practical application are considered. By discussing the uniform distributive load condition, a compromise result is achieved. By discussing the point load condition, the removal work of support structures after printing is alleviated. By discussing the most unfavorable load condition, the worst collapse situation of the printing model during printing process is sufficiently considered. Numerical examples show feasibility and effectiveness of the algorithm.Research limitations/implicationsThe proposed algorithm involves time-consuming finite element analysis and iterative solution, which increase the computation burden. Only the overhang constraint and the minimum compliance problem are discussed, while other constraints and objective functions may be of interest.Practical implicationsCompared with most of the existing heuristic or geometry-based support-generating algorithms, the proposed algorithm develops support structures for AM from a mechanical perspective, which is necessary for support structures particularly used in AM for mega-scale construction such as architectures and sculptures to ensure printing success and accuracy of the printed model.Social implicationsWith the rapid development of AM, complicated structures result from topology optimization are available for fabrication. The present paper demonstrates a combination of AM and topology optimization, which is the trend of fabricating manner in the future.Originality/valueThis paper remarks the first of attempts to use continuum topology optimization method to generate support structures for AM. The methodology used in this work is theoretically meaningful and conclusions drawn in this paper can be of important instruction value and practical significance.

Topology optimization of reinforced concrete structures considering control of shrinkage and strength failure

To take into account the shrinkage effect in the early stage of Reinforced Concrete (RC) design, an effective continuum topology optimization method is presented in this paper. Based on the power-law interpolation, shrinkage of concrete is numerically simulated by introducing an additional design-dependent force. Under multi-axial stress conditions, the concrete failure surface is well fitted by two Drucker–Prager yield functions. The optimization problem aims at minimizing the cost function under yield strength constraints on concrete elements and a structural shrinkage volume constraint. In conjunction with the adjoint-variable sensitivity information, the enhanced aggregation method is utilized to efficiently reduce the computational effort arisen from large-scale strength constraints. Numerical results reveal that the proposed approach can produce a reasonable solution with the least steel reinforcements to ensure the structural safety under the combined action of external loads and shrinkage.

Design of single-axis flexure hinges using continuum topology optimization method

The design of compliant hinges has been extensively studied in the size and shape level in the literature. This paper presents a method for designing the single-axis flexure hinges in the topology level. Two kinds of hinges, that is, the translational hinge and the revolute hinge, are studied. The basic optimization models are developed for topology optimization of the translational hinge and the revolute hinge, respectively. The objective for topology optimization of flexure hinges is to maximize the compliance in the desired direction meanwhile minimizing the compliances in the other directions. The constraints for accomplishing the translational and revolute requirements are developed. The popular Solid Isotropic Material with Penalization method is used to find the optimal flexure hinge topology within a given design domain. Numerical results are performed to illustrate the validity of the proposed method.

A performance-based optimization method for topology design of continuum structures with mean compliance constraints

A performance-based optimization (PBO) method for optimal topology design of linear elastic continuum structures with mean compliance constraints is presented in this paper. The performance-based design concept is incorporated in continuum topology optimization, which is treated as the problem of improving the performance of a continuum design domain in terms of the efficiency of material usage and overall stiffness. A simple scheme is employed in the proposed method to suppress the formation of checkerboard patterns. Two energy-based performance indices are derived for quantifying the topology performance of plane stress structures and plates in bending. Performance-based optimality criteria incorporating performance indices are proposed, and can be used in any continuum topology optimization methods for compliance minimization problems to obtain the optimum. Numerical examples are provided to demonstrate the effectiveness and validity of the PBO method in producing optimal topologies of continuum structures.

Optimal topology selection of continuum structures with displacement constraints

This paper deals with the optimal topology selection of continuum structures subject to displacement constraints by using the performance-based design concept. The optimal topology of a continuum structure is generated by gradually eliminating underutilized elements from the discretized design domain. A performance index is developed for monitoring the optimization process and is used as a termination criterion in the optimization algorithm so that the global optimum can be selected from the optimization history. Maximizing the performance index in the design space is proposed as the performance-based optimization criterion. The performance index can be utilized to compare the efficiency of structural topologies produced by different continuum topology optimization methods. Several examples are provided to demonstrate the capabilities of the performance-based optimization approach in selecting the best configuration for the minimum-weight design of continuum structures with maximum stiffness.