Evolutionary level set method for structural topology optimization

This paper proposes an evolutionary accelerated computational level set algorithm for structure topology optimization. It integrates the merits of evolutionary structure optimization (ESO) and level set method (LSM). Traditional LSM algorithm is largely dependent on the initial guess topology. The proposed method combines the merits of ESO techniques with those of LSM algorithm, while allowing new holes to be automatically generated in low strain energy within the nodal neighboring region during optimization. The validity and robustness of the new algorithm are supported by some widely used benchmark examples in topology optimization. Numerical computations show that optimization convergence is accelerated effectively.     

A semi-Lagrangian level set method for structural optimization

In level set based structural optimization, semi-Lagrange method has an advantage to allow for a large time step without the limitation of Courant–Friedrichs–Lewy (CFL) condition for numerical stability. In this paper, a line search algorithm and a sensitivity modulation scheme are introduced for the semi-Lagrange method. The line search attempts to adaptively determine an appropriate time step in each iteration of optimization. With consideration of some practical characteristics of the topology optimization process, incorporating the line search into semi-Lagrange optimization method can yield fewer design iterations and thus improve the overall computational efficiency. The sensitivity modulation is inspired from the conjugate gradient method in finite-dimensions, and provides an alternative to the standard steepest descent search in level set based optimization. Two benchmark examples are presented to compare the sensitivity modulation and the steepest descent techniques with and without the line search respectively.     

Topology optimization study of arterial bypass configurations using the level set method

We studied the arterial bypass design problem using a level set based topology optimization method. The blood flow in the artery was considered as the non-Newtonian flow governed by the Navier–Stokes equations coupled with the modified Cross model for the shear dependent viscosity. The fluid–solid interface is immersed in the design domain by the level set method and the fictitious porous material method. The sensitivity velocity derived by the level set based continuous adjoint method was utilized to control the evolution of the level set function. In order to accommodate the irregular analysis domains, the flow equations and the level set equations were computed on two different unstructured grids respectively. Three idealized arterial bypass configurations problems with the minimum flow shear stress objective were studied in the numerical examples. The results indicated that the optimal arterial bypass designs can effectively reduce integral of the squared shear rate in the artery and have a superior performance for the arterial grafting.     

Eigenvalue topology optimization of structures using a parameterized level set method

Preventing a structure from resonance is important in many real-world applications. Because an external excitation frequency can be lower than the fundamental eigenfrequency or between the eigenfrequencies of a structure, there is a strong need for eigenfrequency optimization technology to optimize the fundamental eigenfrequency and, in addition, the k-th eigenfrequency and to maximize the gap between eigenfrequencies. However, previous optimization studies on vibrating elastic structures that used the level set method have been devoted to the optimization of the fundamental eigenfrequency, whereas the higher-order eigenfrequencies optimization problem has seldom been considered. This paper presents an eigenfrequency optimization technology that is based on the compactly supported radial basis functions (CS-RBFs) parameterized level-set method, using the fundamental eigenfrequency, the eigenfrequency of a given higher-order, and the gap between two consecutive eigenfrequencies as the optimization objectives. Furthermore, to address the oscillation problem of the objective function, we adopt an exponential weighted optimization model of a number of the lower eigenfrequencies for multiple eigenvalue optimizations, and we utilize mode-tracking technology for the single eigenvalue optimization.In addition, we further extend the CS-RBFs parameterized level-set method to an optimization that is performed with geometric constraints, which means that the size and position of the regular holes in the structure can be optimized with the shape and topology. This approach is useful in real-world applications. The effectiveness of this method is demonstrated by several widely investigated examples that have various objectives.     

Topological shape optimization of geometrically nonlinear structures using level set method

Using the level set method, a topological shape optimization method is developed for geometrically nonlinear structures in total Lagrangian formulation. The structural boundaries are implicitly represented by the level set function, obtainable from “Hamilton-Jacobi type” equation with “up-wind scheme,” embedded into a fixed initial domain. The method minimizes the compliance through the variations of implicit boundary, satisfying an allowable volume requirement. The required velocity field to solve the Hamilton-Jacobi equation is determined by the descent direction of Lagrangian derived from an optimality condition. Since the homogeneous material property and implicit boundary are utilized, the convergence difficulty is significantly relieved.     

A level set method for structural shape and topology optimization using radial basis functions

This paper presents an alternative level set method for shape and topology optimization of continuum structures. An implicit free boundary representation model is established by embedding structural boundary into the zero level set of a higher-dimensional level set function. An explicit parameterization scheme for the level set surface is proposed by using radial basis functions with compact support. In doing so, the originally more difficult shape and topology optimization, driven by the temporal and spatial Hamilton–Jacobi partial differential equation (PDE), is transformed into a relatively easier size optimization of the expansion coefficients of the basis functions. The design optimization is converted to an iterative numerical process that combines the parameterization with a derivation of the shape sensitivity of the design functions, so as to allow using mathematical programming algorithms to solve the level set-based design problem and avoid directly solving the Hamilton–Jacobi PDE. Furthermore, a numerically more stable and efficient volume integration scheme is proposed to implement calculations of the shape derivatives, leading to the creation of new holes which are generated initially along the boundary and then propagated to the interior of the design domain. Two widely studied examples are used to demonstrate the effectiveness of the proposed optimization method.     

A level set based method for the optimization of cast part

A cast part is formed via casting process in which molten liquid is poured into and solidifies in a cavity enclosed by molds. Then, one obtains the cast part when the molds are removed. An important issue in the casting process is that a cast part should have a proper geometry so that the molds can actually be removed. Accordingly, in the optimization of a cast part one not only needs to optimize the performance of the cast part but also needs to ensure the cast part have a proper geometry. With these goals, a level set based method is proposed for the optimization of cast part. A molding condition and a performance condition on the design velocity are derived for the optimization. Numerical examples are provided in 2D and 3D.     

Simultaneous optimization of cast part and parting direction using level set method

Parting direction is one of the main parameters that significantly affect mouldability and manufacturing costs of a cast part. In conventional optimal design of cast part, a parting direction is pre-selected by a designer and fixed throughout the optimization. However, when the optimization is performed with a different parting direction, the resulting design will also be different, and more importantly it will end up with different working performance. Therefore, we take the parting direction as a design variable in the optimization of a cast part so that the working performance can be optimized as much as possible. With these goals, a level set based method is proposed for the simultaneous optimization of cast part and parting direction. In each iteration, an optimal parting direction is first computed for the current structure, then the boundary of the current structure is updated by a design velocity that guarantees the design be moldable with the optimal parting direction. Therefore, although the parting direction may be changed during the optimization, the structure will always be moldable in the current parting direction. Numerical examples are provided in 3D.     

A filter-based level set topology optimization method using a 62-line MATLAB code

Structural and Multidisciplinary Optimization - The present paper proposes a fast and easy to implement level set topology optimization method that is able to adjust the complexity of resulting...