Proportional Topology Optimization: A New Non-Sensitivity Method for Solving Stress Constrained and Minimum Compliance Problems and Its Implementation in MATLAB

Abstract

A new topology optimization method called the Proportional Topology Optimization (PTO) is presented. As a non-sensitivity method, PTO is simple to understand, easy to implement, and is also efficient and accurate at the same time. It is implemented into two MATLAB programs to solve the stress constrained and minimum compliance problems. Descriptions of the algorithm and computer programs are provided in detail. The method is applied to solve three numerical examples for both types of problems. The method shows comparable efficiency and accuracy with an existing optimality criteria method which computes sensitivities. Also, the PTO stress constrained algorithm and minimum compliance algorithm are compared by feeding output from one algorithm to the other in an alternative manner, where the former yields lower maximum stress and volume fraction but higher compliance compared to the latter. Advantages and disadvantages of the proposed method and future works are discussed. The computer programs are self-contained and publicly shared in the website www.ptomethod.org.

Hip Implant Design With Three-Dimensional Porous Architecture of Optimized Graded Density

“…Finite element (FE) analysis (ANSYS, Canonsburg, PA) is used to retrieve the mechanical properties of the implant at each iteration. The PTO [37] is here extended to solve a problem where constraints of pore size, porosity, and strut thickness are converted into lower and upper bounds of element density, with two density-update strategies constructed ad hoc to handle the interface failure and fatigue constraints. The interface failure constraint is also introduced to prevent bone-implant micromotion, and the fatigue constraint is applied to the microlattice structure to guarantee the strength necessary to sustain the physiological set of mechanical loadings.…”

“…Design Variables. The PTO, a simple and efficient nonsensitivity method, has been recently used to solve the classical minimum compliance problem and the stress constrained problem for a solid material [37]. In this paper, the PTO is extended to solve maximum compliance problem of a lattice domain under multiple constraints including interface failure and fatigue, two constraints that were not previously examined [37].…”

“…This paper examines a fully three-dimensional (3D) domain of the implanted femur subjected to multiple design constraints, and presents a time-efficient computational scheme to optimally grade the spatial distribution of its relative density. To solve the largescale 3D problem, we use a recently introduced scheme for topology optimization, the proportional topology optimization (PTO) [37], to minimize one objective function, i.e., bone resorption, and convert the other, i.e., bone-implant interface failure, into a design constraint that prevents bone-implant micromotion. PTO is a gradient-free method chosen here over other stochastic and gradient-based methods, such as solid isotropic material with penalization SIMP [38][39][40][41] and level set [42][43][44][45], for its simplicity and efficiency.…”

Optimization on mechanical structure for material nonlinearity based on proportional topology method

“…Both SIMP and level set methods are a common approach to seek an optimal layout of the structure by requiring sensitivity analysis and level set function for updating the design variable in each iteration, respectively. Stress limitation for linear analysis (yield stress) was applied to be an optimization constraint subject to minimize structural weight by employing SIMP approach with sensitivity [6] and non-sensitivity analysis [7]. The problems under stress and volume constraints acquired the quite different optimal layout.…”

“…A non-sensitivity method based on topology optimization, so-called proportional topology optimization (PTO) was introduced to update the value of each design variable. This method was investigated under the problem of stress constraint and compliance minimization, and the optimal layout of each problem was compared to the topology optimization with optimality criteria method (OC) [20]. It is found that the PTO method is an efficient algorithm to acquire the final results under topology optimization and more facile to implement the algorithm than the OC method.…”