Discrete Material Optimization Research Articles

Multi-material and thickness optimization of laminated composite structures subject to high-cycle fatigue

Laminated composite structures have a distinct inherent potential for optimization due to their tailorability and their associated complex failure mechanisms that makes intuitive design remarkably difficult. Optimization of such is a maturing technology with many criteria and manufacturing constraints having been successfully demonstrated. An approach for high-cycle fatigue is however yet to be developed in a gradient-based context. Thus, the objective of this work is to introduce a novel framework that allows for effective high-cycle fatigue optimization of laminated composite structures.Offset is taken in the Discrete Material and Thickness Optimization parametrization, which allows for simultaneous material and thickness selection for each layer that constitute a laminate. The fatigue analysis approach is based on accumulating damage from all variable-amplitude cycles in an arbitrary spectrum. As high-cycle fatigue behavior is highly nonlinear, it is difficult to handle in optimization. To stabilize the problem, damage is scaled using an inverse P-mean norm formulation that reduces the nonlinearity and provides an accurate measure of the damage. These scaled damages are then aggregated using P-norm functions to reduce the number of constraints. This is convenient, as it allows sensitivities to be efficiently calculated using analytical adjoint design sensitivity analysis. The effectiveness of this approach will be demonstrated on both benchmark examples and a more complicated main spar structure.

Multi-material isogeometric topology optimization in multiple NURBS patches

In design optimization of structures, design domains are often described by multiple non-uniform rational B-spline (NURBS) patches. To perform multi-material topological design in such design domains, an effective Nitsche-based multi-material isogeometric topology optimization (ITO) method is proposed in this paper. Specifically, the Nitsche's method is utilized to effectively couple the interfaces between adjacent NURBS patches. In order to preserve the coercivity of the bilinear form, the stabilization parameters are estimated locally according to the element size. Meanwhile, the discrete material optimization (DMO) approach is employed to describe the mixing of candidate materials. Analytical computation of sensitivities for coupled elements is developed as well. The effectiveness of the proposed method is demonstrated via several 2D and 3D numerical examples, where the design domains are constituted by conforming and non-conforming NURBS patches, respectively.

Concurrent multi-material and multi-scale design optimization of fiber-reinforced composite material and structures for minimum structural compliance

With the development of additive manufacturing technology for fiber-reinforced composite materials, topology optimization of fiber-reinforced composite laminates involving multiple materials and variable stiffness is gaining increasing attention. This study proposes an effective methodology for a Multi-scale and Multi-material Composite Anisotropic Penalization (MMCAP) model to investigate multi-scale and multi-material design optimization of a fiber-reinforced variable stiffness (VS) composite structure to minimize structural compliance. In the concurrent MMCAP model, the macroscopic multi-material structural topology and microscopic discrete fiber laying angle selection are introduced as independent design variables and optimized simultaneously. The modified Solid Isotropic Material with Penalization (SIMP) and the Discrete Material Optimization (DMO) approaches are utilized at the macro- and micro-scales, respectively, to realize a clear macroscopic multi-material structural topology and microscopic specific discrete fiber laying angle selection. Multi-material fiber-reinforced plastic (FRP) materials, such as carbon fiber–reinforced plastic (CFRP) and glass fiber–reinforced plastic (GFRP), are considered two types of solid materials in terms of structural volume cost. Sensitivity analysis of the structural compliance concerning the variables of the two geometrical scales is performed using the analytical sensitivity analysis method. The DMO approach is utilized to couple two geometrical scales: macroscopic topology and microscopic material selection. The capabilities of the proposed MMCAP are demonstrated by concurrent multi-material and multi-scale design optimization of composite panels. The influence of the number of discrete fiber laying angles on the structural compliance and optimized topology configuration is also been discussed. Numerical studies showed that the proposed MMCAP scheme can effectively realize multi-material and multi-scale design optimization of fiber-reinforced composite structure with achieving a clear macroscopic multi-material structural topology and microscopic fiber laying angle. The proposed MMCAP scheme provides a new implementation strategy for lightweight, multi-material, and multi-scale design optimization of composite materials, considering the design and manufacturing collaboration through additive manufacturing technology.

Lightweight Design Method for Three-Wheeled Mobility Frame Using Discrete Material and Thickness Optimization Technique

Lightweight Design Method for Three-Wheeled Mobility Frame Using Discrete Material and Thickness Optimization Technique

A generalized discrete fiber angle optimization method for composite structures: Bipartite interpolation optimization

AbstractThis paper proposes a novel fiber angle optimization method for composite, termed bipartite interpolation optimization (BIO), to address high‐dimensional design variables and inefficient fiber orientation optimization. The weighting functions of BIO are constructed with the help of multiphase material topology optimization approach. Various permutations and combinations of n design variables are utilized to represent 2n discrete candidate angles. Since the weighting functions of the way automatically satisfy the sum constraint, it is unnecessary to address the sum constraints during the optimization process. Compared with the traditional discrete material optimization strategy and solid isotropic material with penalization scheme, the BIO method reduces the dimension of the mathematical model for optimizing fiber angles. Compared to the shape functions with penalization scheme, the BIO method can be extended to the optimization problem of 2n candidates. Numerical examples of different types demonstrate that the proposed method has higher solution efficiency in the optimization problem of fiber orientation selection and is suitable for three‐dimensional shell optimization problems. It provides a new technical means for the structural optimization design of composite laminates.

Performance optimization and broadband design of piezoelectric energy harvesters based on isogeometric topology optimization framework

Energy harvesting is attracting considerable attention in devices that cannot replace exhausted batteries, where improving the energy harvesting efficiency of piezoelectric energy harvesters (PEHs) through different optimization methods is always a research hotspot. This paper proposes an isogeometric topology optimization framework to implement the simultaneous optimization of fiber angles and piezoelectric material and reveal the feasibility of broadband design for PEHs. We deduce the isogeometric formulation of PEHs based on Kirchhoff plate theory and Hamilton's principle. The optimization formulations are established with minimum energy factor or maximum output power considered, and the distributions of fiber angles and piezoelectric material are respectively defined by the Heaviside Penalization of Discrete Material Optimization (HPDMO) model and the Piezoelectric Material with Penalization (PEMAP) model. The numerical results show that there is only a slight difference in the optimized designs of fiber angles between the static load and harmonic load when the objective function is the minimum energy factor. Furthermore, taking maximum output power as the objective function, resistance has a substantial effect on the optimized distributions of fiber angles and piezoelectric material, and there is an optimal resistance making the output power of the optimized design increase 9 times compared with the initial model. More important, by combining the designability of composite material, the proposed method is extended to the broadband design of PEHs. It is concluded that the optimized design can not only generate two resonance peaks in the predetermined frequency range but also the second-order output power amplitude is 14 times higher than that of the benchmark model.

Simultaneous optimization of materials and fiber angles on laminated composite shells for reducing transient sound radiation

Researchers generally study optimization problems against noise using frequency-domain analysis. Nonetheless, ubiquitous transient behaviors indicate that time-domain optimization approaches cannot be ignored. Thus, this paper investigates a simultaneous optimization performed on fiber, material, and layer scales to reduce the transient noise from laminated shells with variable stiffness design. The optimization goal is to minimize the integral of the squared sound pressure from structures over a time interval of interest; the Heaviside Penalization of Discrete Material Optimization (HPDMO) model is employed to choose an adequate fiber angle or soft material for each element and determine its stacking sequence. The transient sound pressure is predicted by the finite element/time-domain boundary element method (FEM/TDBEM). Considering that vibration response on each boundary node needs to be known beforehand in TDBEM, an efficient combination scheme is proposed to derive the sensitivity formulation. Namely, the direct method is applied to obtain the dynamic response sensitivities of all nodes at once; and the results are input into the sensitivity calculation of transient sound radiation addressed by using the adjoint variable method. Numerical examples discuss the influences of the number of candidate fiber angles, usage of soft material, size of patches, selection of objective function, eccentric excitation, asymmetric layers, and geometric feature on optimal design.

Simultaneous topology and fiber path optimization of composite structures with MAC constraints

The pursuit of stiffer lightweight structures has led to the development of new concepts such as the Variable Stiffness Composite Laminates (VSCL), which further enhance the design potential of these highly tailored materials. As the number of design variables rise, more robust topology optimization formulations are necessary to achieve optimal solutions. In this work, a design optimization framework is developed by assigning material and fiber orientation on discrete patches of the VSCL structures. A gradient-based optimization method is used in the framework by applying the Discrete Material Optimization (DMO) formulation. The main goal is to design structures to meet dynamic performance objectives, with particular emphasis on the synthesis of modes with a target design. Accordingly, the objective function depends on the eigenvalues (natural frequencies), while one set of constraints depends on the eigenvectors (mode shapes). The Modal Assurance Criterion (MAC) is simultaneously used to perform mode tracking between the different design iterations and to evaluate the correlation between the synthetized modes and the pre-defined objective shapes. In this study, the calculation of the weight derivatives is also revised for the DMO formulation to show the contribution of the missing terms in the final optimal results. The optimization framework is examined for some potential applications related to dynamic decoupling and dynamic synthesis. It is shown that the design approach tailors the dynamic response of the composite structure and it can effectively be used to design resonators based on a reference model.

Concurrent topology and fiber orientation optimization method for fiber-reinforced composites based on composite additive manufacturing

Additive manufacturing of fiber-reinforced composite (FRC) parts provides a new rapid prototyping technique for designs. However, restricted by the layer-by-layer style of additive manufacturing techniques, the designed fiber orientations should be parallel to the print plane, and that will result in a reduced design space for the orientation optimization. To enlarge the orientation design space under this constraint, this paper proposes a new optimization method for FRC parts. The optimized structure is composed of several groups of components, and the fiber orientation of each group of components is constrained to a specific plane. Based on the composite additive manufacturing technology, these components can be fabricated separately and joined together to form the whole structure. The number of the component groups (also the fiber print planes) is predefined. In this method, the structure is optimized by concurrently designing the topology, the components partitioning and the fiber orientations. The structural topology and components partitioning are described by the SIMP interpolation scheme and discrete material optimization (DMO) interpolation scheme, respectively. The discrete-continuous parameterization (DCP) method with two sub-intervals is applied in the orientation optimization to avoid the local optimum of the angle variables, and a unified formulation is given for the concurrent optimization design. Furthermore, a density projection strategy with self-adaptive threshold parameters for multi-material optimization problems is proposed to get the distinct component interfaces. A stable convergence can be obtained without manually adjusting the projection parameters during the optimization iterations. The corresponding angle filtering method is developed to smooth the optimized fiber orientations. Numerical examples and manufacturing experiments are presented to demonstrate the effectiveness of the proposed method.

Discrete material optimization of composite structures subjected to initial excitation for minimum residual vibration

This work studies design optimization of laminated structures subjected to initial excitation for minimum residual vibration. The Discrete Material Optimization (DMO) method with solid isotropic material with penalization (SIMP) model is employed to perform the design optimization of stacking sequence, fiber angles, and selection of material for the composite structures. The transient vibration analysis of the composite structures excited by initial excitation can be performed via the finite element method, where the residual vibration is identified. The integration of squared displacement of residual vibration can be selected as the objective function, which can be greatly simplified by Lyapunov equation to reduce the computational time. A modal reduction method is employed to further improve the computational efficiency of vibration analysis. Since many linear constraints are included in the optimization formulation for discrete material optimization, sequential linear programming (SLP) algorithm with a trust region approach is used to conduct the optimization iteration with the help of the design sensitivities obtained by an adjoint method. Numerical examples are presented to illustrate the effectiveness of the proposed method. The influences of the distribution and magnitude of several initial excitations on the optimized solutions are discussed.

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

Multi-material topology optimization considering joint stiffness using a two-step filtering approach

In this paper, a new method is presented to concurrently determine the structural layout and joint interface during the multi-material topology optimization (MMTO) process. Although the development of additive manufacturing techniques allows the fabrication of multi-material structures for soft materials with graded properties, joint materials for joining metals or composites are still needed. This paper proposes a novel material interpolation scheme for defining the joint area between dissimilar materials using a two-step filtering process. The general MMTO process is performed in the first filtering step. Furthermore, the filtered variables generate the joint area, and the filter radius controls the thickness of the area in the second filtering step. A modified discrete material optimization (DMO) approach is developed to control the different materials independently and to expand the applicability of the method to cases with more than two design materials. To demonstrate the performance of the proposed method, a compliance minimization problem is formulated for various volume constraints, joint thicknesses, material properties, mesh types, and number of materials. To show the scalability, 3-D design and compliant mechanism design examples are adopted. Based on numerical examples, it was confirmed that the proposed method performs well in various cases; moreover, the results demonstrate that the concurrent designing of the structural layout and joint interface leads to better performance than when joint stiffness is not considered.

Machine learning based topology optimization of fiber orientation for variable stiffness composite structures

AbstractThis study proposes a machine learning (ML) based approach for optimizing fiber orientations of variable stiffness carbon fiber reinforced plastic (CFRP) structures, where neural networks are developed to estimate the objective function and analytical sensitivities with respect to design variables as a substitute for finite element analysis (FEA). To reduce the number of training samples and improve the regression accuracy, an active learning strategy is implemented by successively supplying effective samples along with the suboptimal process. After proper training of neural networks, a quasi‐global search strategy can be applied by implementing a large number of initial designs as starting points in the optimization. In this article, a mathematical example is first presented to show the superiority of the active learning strategy. Then a benchmark design example of a CFRP plate is scrutinized to compare the proposed ML‐based with the conventional FEA‐based discrete material optimization (DMO) method. Finally, topology optimization of fiber orientations is performed for design of a CFRP engine hood, in which the structural performance generated from the proposed ML‐based approach achieves 12.62% improvement compared with that obtained from the conventional single‐initial design method. This article is anticipated to demonstrate a new alternative for design of fiber‐reinforced composite structures.

A new method for simultaneous material and topology optimization of composite laminate structures using Hyperbolic Function Parametrization

This paper presents a new discrete parametrization method for simultaneous topology and material optimization of composite laminate structures, referred to as Hyperbolic Function Parametrization (HFP). The novelty of HFP is the way the candidate materials are parametrized in the optimization problem. In HFP, a filtering technique based on hyperbolic functions is used, such that only one design variable is used for any given number of material candidates. Compared to state-of-the-art methods such Discrete Material and Topology Optimization (DMTO) and Shape Function with Penalization (SFP), HFP has much fewer optimization variables and constraints but introduces additional non-linearity in the optimization problems. A comparative analysis of HFP, DMTO and SFP are performed based on the problem of maximizing the stiffness of composite plates under a total volume constraint and multiple manufacturing constraints using various loads, boundary conditions and input parameters. The comparison shows that all three methods are highly sensitive to the choice of input parameters for the optimization problem, although the performance of HFP is overall more consistent. HFP method performs similarly to DMTO and SFP in terms of the designs obtained and computational cost. However, HFP obtains similar or better objective function values compared to the DMTO and SFP methods.

Discrete material optimization with sandwich failure constraints

Discrete material optimization (DMO) is a method, which was originally developed for designing composite structures via multi-material topology optimization principles. Current study applies DMO to sandwich structures with variable thickness in the core and face sheets. Each layer contains design variables for available materials. Materials are combined through interpolation schemes to define properties of the layer. The objective function (mass of the structure) and the failure constraints are interpolated via Rational Approximation of Material Properties (RAMP) in order to calculate with smooth variables, but achieve discrete results. This enables gradient optimization via Interior Point Optimizer (IPOPT) with constraints on maximum stress, wrinkling, and crimping. Structure is modeled by the finite element method, which calculates element forces and moments repeatedly as the stiffness of the structure changes during optimization. Element loads are used by the first-order shear deformation theory to evaluate the stresses in the layers to obtain failure constraints requested in each iteration by the gradient optimizer. Solution is demonstrated on the plate examples showing material distribution and discreteness level. In addition, constraint aggregation by Kreisselmeier-Steinhauser (KS) function was utilized to decrease the number of constraints in the optimization.

Strength Optimization of Infant Pop-Up Seat Frame Using Discrete Material and Thickness Optimization

In this paper, the authors proposed an optimal design method for the strength design of infant pop-up seat frame combined with rear seats for infants, children, and adults, not removable booster seats or car seats. Frame strength design was performed using discrete material and thickness optimization (DMTO) method considering high strength steel (HSS) and advanced high strength steel (AHSS). Structural design using the Section 4 link mechanism was performed, and the weakness of the seat frame due to static load was confirmed through finite element analysis. An optimal design criterion was established by carrying out a case study to derive the limiting conditions according to static and dynamic loads. In consideration of these criteria, the optimal design according to d-optimal and discrete Latin-hypercube (DLH) was performed among the design of experiments (DOE). And the strength of the pop-up seat frame for infants according to each DOE was checked, and the strength optimization method was suggested by comparing the lightweight ratio.

Multicomponent topology optimization of functionally graded lattice structures with bulk solid interfaces

AbstractThis article presents a topology optimization method for structures consisting of multiple lattice components under a certain size, which can be manufactured with an additive manufacturing machine with a size limit and assembled via conventional joining processes, such as welding, gluing, riveting, and bolting. The proposed method can simultaneously optimize overall structural topology, partitioning to multiple components and functionally graded lattices within each component. The functionally graded lattice infill with guaranteed connectivity is realized by applying the Helmholtz PDE filter with a variable radius on the density field in the solid isotropic material with penalization (SIMP) method. The partitioning of an overall structure into multiple components is realized by applying the discrete material optimization (DMO) method, in which each material is interpreted as each component, and the size limit for each component imposed by a chosen additive manufacturing machine. A gradient‐free coating filter realizes bulk solid boundaries for each component, which provide continuous mating surfaces between adjacent components to enable the subsequent joining. The structural interfaces between the bulk solid boundaries are extracted and assigned a distinct material property, which model the joints between the adjacent components. Several numeral examples are solved for demonstration.

A multi-scale discrete material optimization model for optimization of structural topology and material orientations to minimize dynamic compliance

For fiber-reinforced composites, the combination of structural topology optimization and fiber angle design is an excellent way to suppress structural vibration more efficiently and acquire a lighter structure. This paper proposes a concurrent optimization model for fiber-reinforced composite structures, where the mathematical model is built by combining the polynomial interpolation scheme (PIS) and the Heaviside penalization of discrete material optimization (HPDMO). Therein, the former is adopted to avoid the localized mode phenomena due to the mismatch between element stiffness and mass. And the latter is used for multi-candidate fiber orientation design. The objective function is to minimize the integration of the dynamic compliance in a given frequency band subject to macro-volume constraint. In order to ease the heavy computational burden caused by hundreds of frequency steps in each iteration, too many freedom degrees, and discrete design variables in HPDMO, the modal acceleration method is used to obtain a high-precision estimation of displacement response, and a decoupled method suitable for multiphase fiber material sensitivity analysis is also developed. Finally, a modified threshold Heaviside projection is used to obtain a black-and-white design. Several numerical examples are presented to verify the validity and robustness of the proposed model.

On design of carbon fiber reinforced plastic (CFRP) laminated structure with different failure criteria

This study develops a topology optimization approach for design of carbon fiber reinforced plastic (CFRP) laminated components with different failure criteria to reduce the risk of structural failure. The discrete material and thickness optimization (DMTO) method is adopted to parameterize the design variables of thickness and orientation of CFRP composites, which is driven by the Method of Moving Asymptote (MMA) algorithm. A large number of local constraints associated with the failure criteria are aggregated in terms of a p-norm function. Analytical sensitivities are derived with respect to the design variables. In this study, a battery hanging structure in electrical vehicle (EV) is exemplified; and a DMTO based design is prototyped and validated through the in-house experimental tests first. To prevent different failure modes, the Hashin, Hoffman and Tsai-Wu failure criteria are then imposed as the design constraints together with manufacturing requirements in the optimization. A comparative study is performed for the design with and without such failure criteria. The results demonstrate that the maximum failure index of the optimized structure with the Tsai-Wu failure criterion decreases the most by 40%; and follows by the Hashin and Hoffman criteria by 24% and 33%, respectively. Finally, the CFRP structure is also optimized to design a so-called double-double laminates for demonstrating the generality of the proposed method. The study is anticipated to gain in-depth understanding of how the failure criteria would affect the design of fiber reinforced composite structures to ensure structural integrity.

Topology optimization of composite hyperelastic material using SPIMFO-method

Fiber reinforced materials are used in assorted engineering application and for this reason, new additive manufacturing technologies have been developed for this type of materials. With these technologies, it is possible to construct composite structures with different shapes and desired fiber orientation. Therefore, reinforced composite structures can be designed based on an optimized fiber orientation. Because of it, several works propose different methods to tailor fiber directions. Albeit, the determination of fiber optimized orientation is a problem usually subjected to multiple local minima issues, unless discrete material optimization methods are used. In addition, in these works, usually a linear relation between stress and strain is considered, which limits simulations to small displacements, strains and rotations. Recently, a method named SPIMFO has been developed where the angle is considered a continuous variable and the local minima issues are circumvented. Thus, this work proposes to determine the optimized fiber orientation of a fiber reinforced composite structure by using the SPIMFO method with a constitutive equation in fully nonlinear range based on transversely isotropic neo-Hookean model. A new method to measure the fiber continuity named index of average continuity is proposed and implemented. The results obtained by using the proposed method are compared to results obtained by using a discrete model named NDFO-m, which is proposed in a previous work.