Element Connectivity Parameterization Method Research Articles

A Physics-Driven Method for Determining Wheel—Rail Contact Area With Gradient-Based Optimization

AbstractIn this paper, a physics-based method to inversely determine wheel—rail contact area in their lifecycle is proposed by introducing a continuous optimization pipeline including filtering and projection procedures. First, the element connectivity parameterization method is introduced to construct continuous objections with discrete contact pairs and formulate the physics-based optimization model. Second, the radius-based filter equation is employed for smoothing the design variables to improve the numerical stability and the differentiable step function is introduced to project smoothed design variables into 0–1 discrete integer space to ensure the solution of the optimization model yields discrete contact pairs. Finally, the method of moving asymptotes is constructed for iteratively updating wheel—rail contact area by analyzing the sensitivity of relaxed optimization formulation with respect to design variables until the algorithm converged. The experimental result shows the effectiveness of the proposed method to inversely determine the wheel—rail contact points in their lifecycle compared to the line tracing method; to the best of our knowledge, it is the first attempt to consider wheel—rail contact area in lifecycle service with both the measured profile and the predicted profile data by gradient-based optimization method.

A newly developed qp-relaxation method for element connectivity parameterization to achieve stress-based topology optimization for geometrically nonlinear structures

The aim of this work is to present a novel computational approach to employ the stress-based topology optimization method (STOM) to minimize the volume subject to the locally defined stress constraints of a geometrically nonlinear structure in the framework of the element connectivity parameterization (ECP) method. Considering the locally defined stress constraints in topology optimization (TO) is a classic and challenging engineering problem, and successful optimization procedures have recently been developed using the density-based TO method for linear elastic structures. However, no study has yet considered the static failure constraint when using TO for a geometrically nonlinear structure. Therefore, the present study develops a novel computational approach for the STOM for a geometrically nonlinear structure. To successfully optimize a geometrically nonlinear structure, the unstable element issue must be properly addressed, in addition to the stress singularity issue, the existence of a large number of constraints, and the highly nonlinear behavior of the local stress constraints. To effectively resolve these issues, this research adopts the ECP method to interpolate and optimize the connectivities among solid finite elements. Furthermore, we find that a stress singularity issue linked to the local optima issue arises in the ECP method that is different from that of the density-based TO. By investigating the singularity behavior in detail, we develop a new qp-relaxation method that is suitable for the ECP method. To demonstrate the improved capability of the proposed ECP method with the modified qp-relaxation, several two-dimensional TO problems are solved.

Topology optimization for nonlinear dynamic problem with multiple materials and material-dependent boundary condition

This article develops a new optimization scheme for geometrically nonlinear dynamic topology optimization considering multiple materials and material-dependent boundary conditions. In the framework of convectional topology optimization procedures, some major issues must be addressed regarding complicated analysis and optimization formulations for these difficult conditions, as well as the unstable elements. To rigorously resolve these issues, this article develops a new patch stacking method based on our previous contribution (the element connectivity parameterization method (ECP) and the element stacking method). Compared with existing multi-material topology optimization schemes, the two differences in the present scheme are the stacking of multiple patches of the ECP method on the same discretization pixel, and the selection of one patch or no patch among them. To show the potential usage and limitations of the developed optimization method, several topology optimization examples with the above conditions are solved.

A level-set based topology optimization using the element connectivity parameterization method

This contribution presents a novel and versatile approach to geometrically nonlinear topology optimization by combining the level-set method with the element connectivity parameterization method or ECP. The combined advantages of both methods open up the possibility to treat a wide range of optimization problems involving complex physical and/or geometrical nonlinearities in a general and elegant manner. The level-set method features shape optimization on a fixed mesh, leading to intrinsically black-and-white designs. This approach allows a clear description of location and orientation of the interface, whereas topological changes can still be handled easily. A popular concept used in conventional level-set methods is to map the level-set function to volume-fraction design variables for every element of a finite element mesh. The resulting element density variables are then used to scale the Young’s modulus in each element using the Ersatz material approach. In this work we employ a modified material interpolation method, in which the element density variables, based on a per-element integration of a regularized Heaviside operator applied to the level-set function, are used as element connectivity design variables. The resulting crisp boundary topology optimization method exploits the advantages of ECP in the field of complex nonlinearities and eliminates the need for penalization by the implicit level-set description of the design.

하중-변위 관계를 고려한 기하 비선형 구조물의 위상 최적 설계

This paper is concerned with a computational approach for topology optimization of geometrically nonlinear structures following specific load-displacement trajectories. In our previous works, attention was paid to stabilize topology optimization involving large displacement and a method called the element connectivity parameterization was developed. Here, we aimed to extend the element connectivity parameterization method to find an optimal geometrically nonlinear structure yielding a specific loaddisplacement trajectory. In contrast to designing a stiffest structure, the trajectory design problem requires special consideration in topology optimization formulation and solution procedure. Some numerical problems were considered to test the developed element connectivity parameterization based formulation.

Topology optimization by penalty (TOP) method

In traditional structural topology optimization (TO), the material properties of continuum finite elements of fixed form and coupling are varied to find the optimal topology that satisfies the design problem. We develop an alternative, fundamental formulation where the design space search is dependent on the coupling, and the goal of the topology optimization by penalty (TOP) method is to determine the optimal finite element coupling constraints. By this approach, seemingly disparate topology design problems, e.g., the topology design of structural supports, topology optimization for fluid mechanics problems, topology optimization with convective heat transfer, topology optimization by the element connectivity parameterization method, and the optimal layout of structural joints, can be understood as related formulations under a common topology optimization umbrella, and more importantly, this general framework can be applied to new design problems. For example, in modern multibody dynamics synthesis, the geometric form of finite elements of fixed material properties and interconnectivity are varied to find the optimal topology that satisfies the mechanism design problem. The a priori selection of coupling, e.g., by revolute or translational joints, severely limits the design space search. The TOP method addresses this limitation in a novel way. We develop the methodology and apply the TOP method to the diverse design problems discussed above.

Optimal layout design of three‐dimensional geometrically non‐linear structures using the element connectivity parameterization method

AbstractThe topology design optimization of ‘three‐dimensional geometrically‐non‐linear’ continuum structures is still difficult not only because of the size of the problem but also because of the unstable continuum finite elements that arise during the optimization. To overcome these difficulties, the element connectivity parameterization (ECP) method with two implementation formulations is proposed. In ECP, structural layouts are represented by inter‐element connectivity, which is controlled by the stiffness of element‐connecting zero‐length links. Depending on the link location, ECP may be classified as an external ECP (E‐ECP) or an internal ECP (I‐ECP). In this paper, I‐ECP is newly developed to substantially enhance computational efficiency. The main idea in I‐ECP is to reduce system matrix size by eliminating some internal degrees of freedom associated with the links at voxel level. As for ECP implementation with commercial software, E‐ECP, developed earlier for two‐dimensional problems, is easier to use even for three‐dimensional problems because it requires only numerical analysis results for design sensitivity calculation. The characteristics of the I‐ECP and E‐ECP methods are compared, and these methods are validated with numerical examples. Copyright © 2006 John Wiley & Sons, Ltd.

The element connectivity parameterization formulation for the topology design optimization of multiphysics systems

AbstractIn spite of the success of the element‐density‐based topology optimization method in many problems including multiphysics design problems, some numerical difficulties, such as temperature undershooting, still remain. In this work, we develop an element connectivity parameterization (ECP) formulation for the topology optimization of multiphysics problems in order to avoid the numerical difficulties and yield improved results. In the proposed ECP formulation, finite elements discretizing a given design domain are not connected directly, but through sets of one‐dimensional zero‐length links simulating elastic springs, electric or thermal conductors. The discretizing finite elements remain solid during the whole analysis, and the optimal layout is determined by an optimal distribution of the inter‐element connectivity degrees that are controlled by the stiffness values of the links. The detailed procedure for this new formulation for multiphysics problems is presented. Using one‐dimensional heat transfer models, the problem of the element‐density‐based method is explained and the advantage of the ECP method is addressed. Copyright © 2005 John Wiley & Sons, Ltd.

3차원 설계 영역에서의 요소 연결 매개법을 이용한 위상 최적 설계

The objective of this paper is to present the element connectivity parameterization (ECP) for three dimensional problems. In the ECP method, a continuum structure is viewed as discretized finite elements connected by zerolength elastic links whose stiffness values control the degree of inter-element connectivity. The ECP method can effectively avoid the formation of the low-density unstable elements. These elements appear when the standard element density method is used for geometrical nonlinear problems. In this paper, this ECP method developed for two-dimensional problems is expanded to the design of three-dimensional geometrical nonlinear structures. Among others, the automatic procedure converting standard finite element models to the models suitable for the ECP approach is developed and applied for optimization problems defined on general three-dimensional design domains.