Discrete Material Optimization Method Research Articles

A dichotomous angle refinement discrete material optimization method for fiber composite materials

The discrete material optimization method optimizes the distribution of materials by setting candidate materials using a multi-phase material penalty model. Addressing the issue of reduced efficiency in fiber orientation optimization due to a large number of candidate materials, this paper proposes a dichotomous angle refinement discrete material optimization method to enhance solution speed. This method leverages the efficiency of the bisection approach to rapidly refine the angle in discrete optimization results, thereby narrowing the range of candidate fiber orientations. Furthermore, by introducing fiber orientations perpendicular to the optimized orientation, it mitigates potential local optimization issues during the optimization process, thereby enhancing the method’s optimization capability. Compared to traditional discrete material optimization methods, this method employs a special dichotomy method to determine the candidate materials for each element, thereby decreasing the number of design variables. This method demonstrates higher computational efficiency when solving fiber orientation optimization problems with a large number of candidate materials. Several numerical examples provided in the paper validate the efficiency and stability of the proposed method in addressing optimization problems.

A discrete material optimization method with a patch strategy based on stiffness matrix interpolation

A discrete material optimization method with a patch strategy based on stiffness matrix interpolation

Two-scale concurrent topology optimization of lattice structures with connectable microstructures

Two-scale concurrent design of lattice material microstructures and their macroscale distributions can effectively enlarge the design space, and thus achieve lightweight structures with desirable mechanical and multi-physics performances. Most of the existing inverse homogenization-based design methods do not take into consideration the connectivity issues of the microstructures. In practice, this may hinder the manufacturing and application of the optimized two-scale designs. To handle this issue, the present paper proposes a designable connective region method to obtain connectable microstructures in the context of two-scale concurrent structural topology optimization, considering structures composed of lattice materials with repetitive unit cells and prescribed porosity. On the microscale, the microstructures topologies are represented by the density model, and the effective material properties are computed with the homogenization method. On the macroscale, the distribution of different lattice materials is described with the discrete material optimization method, which can effectively reduce the amounts of macroscale elements with intermediate densities. The connectivity between any two types of the microstructures is naturally ensured by introducing pre-defined connective regions in the microstructural unit cells and keeping these regions sharing the same topology. This method can be conveniently implemented through microscale design variable linking and requires no evaluation of extra connectivity constraints and the corresponding sensitivities. It is exemplified by two-scale concurrent structural topology optimization for compliance minimization problems in two- and three-dimensional design domains. Numerical results show that this method is able to generate connectable lattice structures, which exhibits improved stiffness as compared with their uniform-lattice counterparts.

Multiscale eigenfrequency optimization of multimaterial lattice structures based on the asymptotic homogenization method

Ultralight lattice structures exhibit excellent mechanical performance and have been used widely. In structural design, the fundamental frequency is highly important. Therefore, a multiscale topology optimization method was utilized to optimize the fundamental frequency of multimaterial lattice structures in this study. Two types of optimization problems were studied, namely, maximizing the natural fundamental frequency with mass constraints and minimizing compliance with frequency constraints. The Heaviside-penalty-based discrete material optimization method was adopted for the optimal selection of candidate materials. The asymptotic homogenization method was used to evaluate the equivalent macroscale properties according to the microstructure of the lattice material. To enable gradient optimization, sensitivities were outlined in detail. A density filter with a volume-preserving Heaviside projection was used to eliminate the risk of a checkerboard pattern and reduce the number of gray elements. A polynomial penalization scheme was employed to eliminate localized spurious eigenmodes in the low-density region. Finally, several numerical examples were performed to validate the proposed method. These numerical examples resulted in novel microstructural configurations with remarkably improved vibration resistance.

Discrete material optimization of vibrating composite plate and attached piezoelectric fiber composite patch

This work deals with the layout optimization of piezoelectric fiber composite patches on a vibrating laminated composite plate and the discrete material design of the composite plate. The vibration of the composite plate is excited by an external mechanical loading, and a sinusoidal voltage with given amplitude and frequency is applied on the piezoelectric fiber composite patches. The analysis of the composite structure with piezoelectric fiber composite patches is performed via a finite element method in condensed form, where the piezoelectric effects are considered as induced force. As a view to minimize the dynamic response of the vibrating laminated composite structure, the Discrete Material Optimization method is employed to perform the design optimization of piezoelectric fiber composite patches and the stacking sequence, fiber angles, and selection of material for the composite structure. Numerical examples are presented to demonstrate the effectiveness of the proposed method.

Topology Optimization Design of Compliant Mechanism of Composite Wing Leading Edge

This paper introduces a topology optimization design method for flexible mechanism of composite wing leading edge based on discrete material optimization (DMO) with composite material. The DMO method is combined with the topology optimization to select the deviation of the actual displacement and the target displacement of the 10 coordinate points on the initial curve of the leading edge as the objective function, and the volume fraction of each phase material is used as the constraint to establish the composite material optimization model, which is solved by the OC optimization criterion method. The MATLAB codes are compiled to calculate the optimization problem and a distinct result image is obtained. The CATIA software is used to model the topology image in three dimensions, and the model is imported into Hyperworks software for simulation analysis. The results show that this complaint mechanism can achieve a maximum deflection of 9.1 degrees, which verifies the feasibility of the method.

Concurrent topology optimization of multiscale structures with multiple porous materials under random field loading uncertainty

Design of multiscale structures is a challenging task due to a vast design space of both materials and structures. Consideration of load uncertainty adds another level of complexity. In this paper, a robust concurrent TO (topology optimization) approach is developed for designing multiscale structures composed of multiple porous materials under random field loading uncertainty. To determine the optimal distribution of the porous materials at the macro/structural scale, our key idea is to employ the discrete material optimization method to interpolate the material properties for multiple porous materials. In addition, for the first time we interpret the interpolation schemes in the existing concurrent TO model of porous material with a clear physical meaning by putting forward a SIMP-like single interpolation scheme. This scheme integrates the SIMP (Solid Isotropic Material with Penalization) at the microscale and PAMP (Porous Anisotropic Material with Penalization) at the macroscale into a single equation. Efficient uncertainty characterization and propagation methods based on K-L expansion and linear superposition are introduced, and several important improvements in objective function evaluation and sensitivity analysis are presented. Improved sensitivity analysis equations are derived for volume preserving filtering, which is employed to deal with numerical instabilities at the macro and micro scales in the robust concurrent TO model. Measures to ensure manufacturability and to improve analysis accuracy and efficiency are devised. 2D and 3D examples demonstrate the effectiveness of the proposed approach in simultaneously obtaining robust optimal macro structural topology and material microstructural topologies.

Optimal design of laminated piezocomposite energy harvesting devices considering stress constraints

Summary Energy harvesting devices are smart structures capable of converting the mechanical energy (generally, in the form of vibrations) that would be wasted otherwise in the environment into usable electrical energy. Laminated piezoelectric plate and shell structures have been largely used in the design of these devices because of their large generation areas. The design of energy harvesting devices is complex, and they can be efficiently designed by using topology optimization methods (TOM). In this work, the design of laminated piezocomposite energy harvesting devices has been studied using TOM. The energy harvesting performance is improved by maximizing the effective electric power generated by the piezoelectric material, measured at a coupled electric resistor, when subjected to a harmonic excitation. However, harmonic vibrations generate mechanical stress distribution that, depending on the frequency and the amplitude of vibration, may lead to piezoceramic failure. This study advocates using a global stress constraint, which accounts for different failure criteria for different types of materials (isotropic, piezoelectric, and orthotropic). Thus, the electric power is maximized by optimally distributing piezoelectric material, by choosing its polarization sign, and by properly choosing the fiber angles of composite materials to satisfy the global stress constraint. In the TOM formulation, the Piezoelectric Material with Penalization and Polarization material model is applied to distribute piezoelectric material and to choose its polarization sign, and the Discrete Material Optimization method is applied to optimize the composite fiber orientation. The finite element method is adopted to model the structure with a piezoelectric multilayered shell element. Numerical examples are presented to illustrate the proposed methodology. Copyright © 2015 John Wiley & Sons, Ltd.

In-plane material filters for the discrete material optimization method

This paper presents in-plane material filters for the Discrete Material Optimization method used for optimizing laminated composite structures. The filters make it possible for engineers to specify a minimum length scale which governs the minimum size of areas with constant material continuity. Consequently, engineers can target the available production methods, and thereby increase its manufacturability while the optimizer is free to determine which material to apply together with an optimum location, shape, and size of these areas with constant material continuity. By doing so, engineers no longer have to group elements together in so-called patches, so to statically impose a minimum length scale. The proposed method imposes the minimum length scale through a standard density filter known from topology optimization of isotropic materials. This minimum length scale is generally referred to as the filter radius. However, the results show that the density filter alone gives designs with large measures of non-discreteness. In order to obtain near discrete designs an additional threshold projection filter is applied, so to push the physical design variables towards their discrete bounds. However, because the projection filter is a non-linear function of the design variables, the projected variables have to be re-scaled in a final so-called normalization filter. This is done to prevent the optimizer in creating superior, but non-physical pseudo-materials. The method is demonstrated on a series of minimum compliance examples together with a minimum mass example, and the results show that the method is indeed capable of imposing a minimum length scale onto the optimized layup.

Topology and thickness optimization of laminated composites including manufacturing constraints

This paper presents a gradient based optimization method for large-scale topology and thickness optimization of fiber reinforced monolithic laminated composite structures including certain manufacturing constraints to attain industrial relevance. This facilitates application of predefined fiber mats and reduces the risk of failure such as delamination and matrix cracking problems. The method concerns simultaneous determination of the optimum thickness and fiber orientation throughout a laminated structure with fixed outer geometry. The laminate thickness may vary as an integer number of plies, and possible fiber orientations are limited to a finite set. The conceptual combinatorial problem is relaxed to a continuous problem and solved on basis of interpolation schemes with penalization through the so-called Discrete Material Optimization method, explicitly including manufacturing constraints as a large number of sparse linear constraints. The methodology is demonstrated on several numerical examples.

Buckling topology optimization of laminated multi-material composite shell structures

The design problem of maximizing the buckling load factor of laminated multi-material composite shell structures is investigated using the so-called Discrete Material Optimization (DMO) approach. The design optimization method is based on ideas from multi-phase topology optimization where the material stiffness is computed as a weighted sum of candidate materials, thus making it possible to solve discrete optimization problems using gradient based techniques and mathematical programming. The potential of the DMO method to solve the combinatorial problem of proper choice of material and fiber orientation simultaneously is illustrated for multilayered plate examples and a simplified shell model of a spar cap of a wind turbine blade.

Discrete material optimization of general composite shell structures

AbstractA novel method for doing material optimization of general composite laminate shell structures is presented and its capabilities are illustrated with three examples. The method is labelled Discrete Material Optimization (DMO) but uses gradient information combined with mathematical programming to solve a discrete optimization problem. The method can be used to solve the orientation problem of orthotropic materials and the material selection problem as well as problems involving both. The method relies on ideas from multiphase topology optimization to achieve a parametrization which is very general and reduces the risk of obtaining a local optimum solution for the tested configurations. The applicability of the DMO method is demonstrated for fibre angle optimization of a cantilever beam and combined fibre angle and material selection optimization of a four‐point beam bending problem and a doubly curved laminated shell. Copyright © 2005 John Wiley & Sons, Ltd.