Topology optimization of microstructures of viscoelastic damping materials for a prescribed shear modulus

Damping performance of a passive constrained layer damping (PCLD) structure mainly depends on the geometric layout and physical properties of the viscoelastic damping material. Properties such as the shear modulus of the damping material need to be tailored for improving the damping of the structures. This paper presents a topology optimization method for designing the microstructures in 2D, i.e., the structure of the periodic unit cell (PUC), of cellular viscoelastic materials with a prescribed shear modulus. The effective behavior of viscoelastic materials is derived through the use of a finite element based homogenization method. Only isotropic matrix material was considered and under such assumption it is found that the effective loss factor of viscoelastic material is independent of the geometrical configuration of the PUC. Based upon the idea of a Solid Isotropic Material with Penalization (SIMP) method of topology optimization, the relative material densities of the elements of the PUC are considered as the design variables. The topology optimization problem of viscoelastic cellular material with a prescribed property and with constraints on the isotropy and volume fraction is established. The optimization problem is solved using the sequential linear programming (SLP) method. Several examples of the design optimization of viscoelastic cellular materials are presented to demonstrate the validity of the method. The effectiveness of the design method is illustrated by comparing a solid and an optimized cellular viscoelastic material as applied to a cantilever beam with the passive constrained layer damping treatment.     

最小位移类桁架连续体拓扑设计优化

为将拓扑优化中的柔度最小化问题拓展到一般位移最小化问题,用有限元划分设计域,采用类桁架连续体材料模型,并假设杆件在设计域内连续分布.将杆件在节点位置的密度和方向作为设计变量,将指定位置和方向的位移作为目标函数,采用基于目标函数梯度的优化准则法,通过优化杆件的连续分布场形成拓扑优化的类桁架连续体.该方法可结合结构力学的基本概念,选择部分杆件形成拓扑优化刚架.     

Adaptive volume constraint algorithm for stress limit-based topology optimization

Homogenization or density-based topology optimization methods work by distributing a fixed amount of material to the most effective areas of the design domain so as to create an optimal structural configuration that meets the minimum compliance criteria. These topology optimization methods generally cannot control the maximum stress levels of the structure; therefore, the smoothened optimum structure is not guaranteed to be ready for immediate use. This can be because it is either unsafe if the maximum stress at this structure exceeds the strength limit, or over designed if the maximum stress is far below the stress limit. Difficult and complex shape optimization must then be done to obtain a minimum-weight structure that meets the maximum stress constraint. This paper proposes an adaptive volume constraint (AVC) algorithm, a heuristic approach, in place of traditional topology optimization methods so that the maximum stress in the optimal structural configuration will be below the predefined stress limit and the smoothened structure will be directly applicable. In order to test the applicability and robustness of the AVC algorithm, topology optimization using both a traditional fixed volume constraint and an AVC are tested on a number of configuration design problems. To further illustrate the usefulness of the AVC algorithm, shape optimizations at the maximum stress constraint are also conducted on the smooth structural models by both optimization approaches on an identical problem set.     

Eulerian shape design sensitivity analysis and optimization with a fixed grid

Conventional shape optimization based on the finite element method uses Lagrangian representation in which the finite element mesh moves according to shape change, while modern topology optimization uses Eulerian representation. In this paper, an approach to shape optimization using Eulerian representation such that the mesh distortion problem in the conventional approach can be resolved is proposed. A continuum geometric model is defined on the fixed grid of finite elements. An active set of finite elements that defines the discrete domain is determined using a procedure similar to topology optimization, in which each element has a unique shape density. The shape design parameter that is defined on the geometric model is transformed into the corresponding shape density variation of the boundary elements. Using this transformation, it has been shown that the shape design problem can be treated as a parameter design problem, which is a much easier method than the former. A detailed derivation of how the shape design velocity field can be converted into the shape density variation is presented along with sensitivity calculation. Very efficient sensitivity coefficients are calculated by integrating only those elements that belong to the structural boundary. The accuracy of the sensitivity information is compared with that derived by the finite difference method with excellent agreement. Two design optimization problems are presented to show the feasibility of the proposed design approach.     

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.     

Optimal topology design for stress-isolation of soft hyperelastic composite structures under imposed boundary displacements

Soft hyperelastic composite structures that integrate soft hyperelastic material and linear elastic hard material can undergo large deformations while isolating high strain in specified locations to avoid failure. This paper presents an effective topology optimization-based methodology for seeking the optimal united layout of hyperelastic composite structures with prescribed boundary displacements and stress constraints. The optimization problem is modeled based on the power-law interpolation scheme for two candidate materials (one is soft hyperelastic material and the other is linear elastic material). The ?-relaxation technique and the enhanced aggregation method are employed to avoid stress singularity and improve the computational efficiency. Then, the topology optimization problem can be readily solved by a gradient-based mathematical programming algorithm using the adjoint variable sensitivity information. Numerical examples are given to show the importance of considering prescribed boundary displacements in the design of hyperelastic composite structures. Moreover, numerical solutions demonstrate the validity of the present model for the optimal topology design with a stress-isolated region.     

Weight optimization for flexural reinforced concrete beams with static nonlinear response

The weight optimization of reinforced concrete (RC) beams with material nonlinear response is formulated as a general nonlinear optimization problem. Incremental finite element procedures are used to integrate the structural response analysis and design sensitivity analysis in a consistent manner. In the finite element discretization, the concrete is modelled by plane stress elements and steel reinforcement is modelled by discrete truss elements. The cross-sectional areas of the steel and the thickness of the concrete are chosen as design variables, and design constraints can include the displacement, stress and sizing constraints. The objective function is the weight of the RC beams. The optimal design is performed by using the sequential linear programming algorithm for the changing process of design variables, and the gradient projection method for the calculations of the search direction. Three example problems are considered. The first two are demonstrated to show the stability and accuracy of the approaches by comparing previous results for truss and plane stress elements, separately. The last one is an example of an RC beam. Comparative cost objective functions are presented to prove the validity of the approach.     

Adaptive topology optimization

Topology optimization of continuum structures is often reduced to a material distribution problem. Up to now this optimization problem has been solved following a rigid scheme. A design space is parametrized by design patches, which are fixed during the optimization process and are identical to the finite element discretization. The structural layout is determined, whether or not there is material in the design patches. Since many design patches are necessary to describe approximately the structural layout, this procedure leads to a large number of optimization variables. Furthermore, due to a lack of clearness and smoothness, the results obtained can often only be used as a conceptual design idea.To overcome these shortcomings adaptive techniques, which decrease the number of optimization variables and generate smooth results, are introduced. First, the use of pure mesh refinement in topology optimization is discussed. Since this technique still leads to unsatisfactory results, a new method is proposed that adapts the effective design space of each design cycle to the present material distribution. Since the effective design space is approximated by cubic or Bézier splines, this procedure does not only decrease the number of design variables and lead to smooth results, but can be directly joined to conventional shape optimization. With examples for maximum stiffness problems of elastic structures the quality of the proposed techniques is demonstrated.     

PolyTop: a Matlab implementation of a general topology optimization framework using unstructured polygonal finite element meshes

We present an efficient Matlab code for structural topology optimization that includes a general finite element routine based on isoparametric polygonal elements which can be viewed as the extension of linear triangles and bilinear quads. The code also features a modular structure in which the analysis routine and the optimization algorithm are separated from the specific choice of topology optimization formulation. Within this framework, the finite element and sensitivity analysis routines contain no information related to the formulation and thus can be extended, developed and modified independently. We address issues pertaining to the use of unstructured meshes and arbitrary design domains in topology optimization that have received little attention in the literature. Also, as part of our examination of the topology optimization problem, we review the various steps taken in casting the optimal shape problem as a sizing optimization problem. This endeavor allows us to isolate the finite element and geometric analysis parameters and how they are related to the design variables of the discrete optimization problem. The Matlab code is explained in detail and numerical examples are presented to illustrate the capabilities of the code.     

Implementing a structural continuity constraint and a halting method for the topology optimization of energy absorbers

This study investigates topology optimization of energy absorbing structures in which material damage is accounted for in the optimization process. The optimization objective is to design the lightest structures that are able to absorb the required mechanical energy. A structural continuity constraint check is introduced that is able to detect when no feasible load path remains in the finite element model, usually as a result of large scale fracture. This assures that designs do not fail when loaded under the conditions prescribed in the design requirements. This continuity constraint check is automated and requires no intervention from the analyst once the optimization process is initiated. Consequently, the optimization algorithm proceeds towards evolving an energy absorbing structure with the minimum structural mass that is not susceptible to global structural failure. A method is also introduced to determine when the optimization process should halt. The method identifies when the optimization method has plateaued and is no longer likely to provide improved designs if continued for further iterations. This provides the designer with a rational method to determine the necessary time to run the optimization and avoid wasting computational resources on unnecessary iterations. A case study is presented to demonstrate the use of this method.     

Stress-based topology optimization for continua

We propose an effective algorithm to resolve the stress-constrained topology optimization problem. Our procedure combines a density filter for length scale control, the solid isotropic material with penalization (SIMP) to generate black-and-white designs, a SIMP-motivated stress definition to resolve the stress singularity phenomenon, and a global/regional stress measure combined with an adaptive normalization scheme to control the local stress level.     

Topology optimization considering fracture mechanics behaviors at specified locations

As a typical form of material imperfection, cracks generally cannot be avoided and are critical for load bearing capability and integrity of engineering structures. This paper presents a topology optimization method for generating structural layouts that are insensitive/sensitive as required to initial cracks at specified locations. Based on the linear elastic fracture mechanics model (LEFM), the stress intensity of initial cracks in the structure is analyzed by using singularity finite elements positioned at the crack tip to describe the near-tip stress field. In the topology optimization formulation, the J integral, as a criterion for predicting crack opening under certain loading and boundary conditions, is introduced into the objective function to be minimized or maximized. In this context, the adjoint variable sensitivity analysis scheme is derived, which enables the optimization problem to be solved with a gradient-based algorithm. Numerical examples are given to demonstrate effectiveness of the proposed method on generating structures with desired overall stiffness and fracture strength property. This method provides an applicable framework incorporating linear fracture mechanics criteria into topology optimization for conceptual design of crack insensitive or easily detachable structures for particular applications.     

Design of distributed compliant micromechanisms with an implicit free boundary representation

In this paper, a parameterization approach is presented for structural shape and topology optimization of compliant mechanisms using a moving boundary representation. A level set model is developed to implicitly describe the structural boundary by embedding into a scalar function of higher dimension as zero level set. The compactly supported radial basis function of favorable smoothness and accuracy is used to interpolate the level set function. Thus, the temporal and spatial initial value problem is now converted into a time-separable parameterization problem. Accordingly, the more difficult shape and topology optimization of the Hamilton–Jacobi equation is then transferred into a relatively easy size optimization with the expansion coefficients as design variables. The design boundary is therefore advanced by applying the optimality criteria method to iteratively evaluate the size optimization so as to update the level set function in accordance with expansion coefficients of the interpolation. The optimization problem of the compliant mechanism is established by including both the mechanical efficiency as the objective function and the prescribed material usage as the constraint. The design sensitivity analysis is performed by utilizing the shape derivative. It is noted that the present method is not only capable of simultaneously addressing shape fidelity and topology changes with a smooth structural boundary but also able to avoid some of the unfavorable numerical issues such as the Courant–Friedrich–Levy condition, the velocity extension algorithm, and the reinitialization procedure in the conventional level set method. In particular, the present method can generate new holes inside the material domain, which makes the final design less insensitive to the initial guess. The compliant inverter is applied to demonstrate the availability of the present method in the framework of the implicit free boundary representation.     

Combined optimization of bi-material structural layout and voltage distribution for in-plane piezoelectric actuation

This paper investigates the combined optimization of bi-material structural layout and actuation voltage distribution of structures with embedded in-plane piezoelectric actuators. The maximization of the nodal displacement at a selected output port is considered as the design objective. A two-phase material model with power-law penalization is employed in the topology optimization of the actuator elements and the coupled surrounding structure. In order to incorporate the actuation voltage directly into the design for achieving the best overall actuation performance, element-wise voltage design variables are also included in the optimization. For the purpose of easy implementation of the electric system, the allowable voltage levels at an individual element are confined to three discrete values, namely zero and two prescribed values with opposite signs. To this end, a special interpolation scheme between the tri-level voltage values and the design variables is used in the optimization model. Based on the design sensitivity analysis of the objective function, the combined optimization problem is solved with the MMA algorithm. Numerical examples are presented to demonstrate the applicability of the proposed optimization model and numerical techniques. The optimal solutions also confirmed that larger output displacement can be achieved by introducing voltage design variables into the design problem.     

Topology optimization of cellular materials with periodic microstructure under stress constraints

Material design is a critical development area for industries dealing with lightweight construction. Trying to respond to these industrial needs topology optimization has been extended from structural optimization to the design of material microstructures to improve overall structural performance. Traditional formulations based on compliance and volume control result in stiffness-oriented optimal designs. However, strength-oriented designs are crucial in engineering practice. Topology optimization with stress control has been applied mainly to (macro) structures, but here it is applied to material microstructure design. Here, in the context of density-based topology optimization, well-established techniques and analyses are used to address known difficulties of stress control in optimization problems. A convergence analysis is performed and a density filtering technique is used to minimize the risk of results inaccuracy due to coarser finite element meshes associated with highly non-linear stress behavior. A stress-constraint relaxation technique (qp-approach) is applied to overcome the singularity phenomenon. Parallel computing is used to minimize the impact of the local nature of the stress constraints and the finite difference design sensitivities on the overall computational cost of the problem. Finally, several examples test the developed model showing its inherent difficulties.

     

Stress isolation through topology optimization

This paper introduces a problem of stress isolation in structural design and presents an approach to the problem through topology optimization. We model the stress isolation problem as a topology optimization problem with multiple stress constraints in different regions. The shape equilibrium constraint approach is employed to effectively control the local stress constraints. The level set based structural optimization is implemented with the extended finite element method (X-FEM) for providing an adequately accurate stress analysis. Numerical examples of stress isolation design in two dimensions are investigated as a benchmark test of the proposed method. The results, from the force transmittance point of view, suggest that the guard “grooves” obtained can change the force path to successfully realize the stress isolation in the structure.