Material Interpolation Scheme Research Articles

TOP2DFVT: An Efficient Matlab Implementation for Topology Optimization based on the Finite-Volume Theory

The finite-volume theory has shown to be numerically efficient and stable for topology optimization of continuum elastic structures. The significant features of this numerical technique are the local satisfaction of equilibrium equations and the employment of compatibility conditions along edges in a surface-averaged sense. These are essential properties to adequately mitigate some numerical instabilities in the gradient version of topology optimization algorithms, such as checkerboard, mesh dependence, and local minima issues. Several computational tools have been proposed for topology optimization employing analysis domains discretized with essential features for finite-element approaches. However, this is the first contribution to offer a platform to generate optimized topologies by employing a Matlab code based on the finite-volume theory for compliance minimization problems. The Top2DFVT provides a platform to perform 2D topology optimization of structures in Matlab, from domain initialization for structured meshes to data post-processing. This contribution represents a significant advancement over earlier publications on topology optimization based on the finite-volume theory, which needed more efficient computational tools. Moreover, the Top2DFVT algorithm incorporates SIMP and RAMP material interpolation schemes alongside sensitivity and density filtering techniques, culminating in a notably enhanced optimization tool. The application of this algorithm to various illustrative cases confirms its efficacy and underscores its potential for advancing the field of structural optimization.

Toward static and transient stress-constrained topology optimization for shell-infill structures

This paper contributes to a stress-constrained topology optimization approach for the design of shell-infill structures. Based on the density method, we utilize the two-step filtering and projection scheme and local volume constraint for generating the shell and non-uniform infills, respectively. The topology optimization formulation is defined as a Robust Minimum Compliance problem with Volume and Stress constraints (RMCVS), where the robust method is used to eliminate undesired topological characteristics. We globalize the static and transient stress constraints through the p-norm function, and tailor material interpolation and stress relaxation schemes for both cases, respectively, to avoid numerical difficulties. Sensitivity analysis is provided, and the derivatives of the objective function and constraints with respect to the design variable fields are derived. We then validate the suggested method through several 2D and 3D benchmarks. The results illustrate that the method is robust to generate various shell-infill structures while limiting the maximum static and transient stress.

An improved proportional topology optimization method combining a polarized material interpolation scheme and Heaviside threshold function

An improved proportional topology optimization (IPTO) method is proposed in this work. The main improvement of this method is that the conventional solid isotropic material with penalization (SIMP)-based material interpolation scheme is replaced by a polarized material interpolation scheme, and the Heaviside threshold function is adopted based on the original proportional topology optimization (PTO) method. By using this approach, the minimum compliance problem can be solved without requiring the numerical derivation of the sensitivity function. To verify the feasibility and effectiveness of the proposed method, two-dimensional (2D) and three-dimensional (3D) cantilevers and L-bracket beams are used as examples. The 2D results obtained by the IPTO method are compared with those obtained by the PTO and SIMP methods. Numerical examples demonstrate that IPTO can acquire better objective function values and more ideal topology structures compared to PTO and SIMP. Furthermore, IPTO offers significant advantages over PTO and SIMP in terms of convergence speed and the ability to suppress intermediate density elements. Additionally, this method enables topology optimization design under multiple working conditions. Therefore, it provides an effective approach for structural topology optimization in research and engineering applications. With appropriate adjustment, this method can also be applied to composite material design and heat conduction design.

Modified proportional topology optimization algorithm for multiple optimization problems

Three modified proportional topology optimization (MPTO) algorithms are presented in this paper, which are named MPTOc, MPTOs and MPTOm, respectively. MPTOc aims to address the minimum compliance problem with volume constraint, MPTOs aims to solve the minimum volume fraction problem under stress constraint, and MPTOm aims to tackle the minimum volume fraction problem under compliance and stress constraints. In order to get rid of the shortcomings of the original proportional topology optimization (PTO) algorithm and improve the comprehensive performance of the PTO algorithm, the proposed algorithms modify the material interpolation scheme and introduce the Heaviside threshold function based on the PTO algorithm. To confirm the effectiveness and superiority of the presented algorithms, multiple optimization problems for the classical MBB beam are solved, and the original PTO algorithm is compared with the new algorithms. Numerical examples show that MPTOc, MPTOs and MPTOm enjoy distinct advantages over the PTO algorithm in the matter of convergence efficiency and the ability to obtain distinct topology structure without redundancy. Moreover, MPTOc has the fastest convergence speed among these algorithms and can acquire the smallest (best) compliance value. In addition, the new algorithms are also superior to PTO concerning suppressing gray-scale elements.

Evolutionary topology optimization approach to design multiphase soundproof systems with poroelastic media

SummaryWith the constant development of cities, noise sources have become increasingly present inside and outside living environments. Consequently, soundproof systems comprised of porous materials have been widely adopted as filling fabric of closed‐space structures, such as in the components of buildings, airplanes or automobiles. However, in many situations, simply filling spaces may not be the most effective approach. In that scenario, this work introduces a multiphase acoustic topology optimization methodology to design closed‐space structures for sound attenuation. Based on the Bi‐directional Evolutionary Structural Optimization (BESO) algorithm, the proposed approach combines Biot's poroelasticity equations, expressed in the mixed /p form, and the Unified Multiphase (UMP) technique to fully describe the multiphysics involved in the acoustic, poroelastic and elastic model relations. An objective function contemplating different combinations of structural, viscous and thermal dissipated powers is maximized over multiple frequencies. Volume constraints in each material phase and a novel material interpolation scheme are also considered. The resultant topologies present enhanced dissipated power levels and manufacturability, even when compared with various baseline configurations of similar volume fractions.

Design of multi-material structures using material jetting technology: Topology optimisation, numerical analysis and experiments

This paper presents a thorough experimental/numerical validation of optimised multi-material structures fabricated by material jetting technology. The proposed methodology uses, on the one hand, non-uniform rational basis spline (NURBS) entities to represent the geometric descriptor associated with each material phase constituting the continuum and, on the other hand, a general multi-phase material interpolation scheme to penalise the stiffness tensor of the structure. Two design requirements are included in the problem formulation: the lightness and the minimum length scale of each material phase. The influence of the integer parameters intervening in the definition of the NURBS entity and the influence of different combinations of material phases on the optimised solutions are investigated. The proposed approach is applied to 2D and 3D benchmark structures subjected to prescribed displacements representative of a three-point bending test. Based on the result of the topology optimisation process one of the optimised solutions, balancing the requirements of structural stiffness, lightness, and manufacturing constraints, is selected, manufactured and tested. A comparison between experimental and numerical results (obtained by non-linear analyses) is carried out to show the effectiveness of the approach.

Topology Optimization of Geometrically Nonlinear Structures Based on a Self-Adaptive Material Interpolation Scheme

Topology Optimization of Geometrically Nonlinear Structures Based on a Self-Adaptive Material Interpolation Scheme

Soft- and Hard-Kill Hybrid GPU-based Bi-Directional Evolutionary Structural Optimization

Abstract Bi-directional evolutionary structural optimization (BESO) is a well-recognized method for generating optimal topologies of structures. Its soft-kill variant has a high computational cost, especially for large-scale structures, whereas the hard-kill variant often faces convergence issues. Addressing these issues, this paper proposes a hybrid BESO model tailored for graphics processing units (GPUs) by combining the soft-kill and hard-kill approaches for large-scale structures. A GPU-based algorithm is presented for dynamically isolating the solid/hard elements from the void/soft elements in the finite element analysis (FEA) stage. The hard-kill approach is used in the FEA stage with an assembly-free solver to facilitate the use of high-resolution meshes without exceeding the GPU memory limits, whereas for the rest of the optimization procedure, the soft-kill approach with a material interpolation scheme is implemented. Furthermore, the entire BESO method pipeline is accelerated for both the proposed hybrid and the standard soft-kill BESO. The comparison of the hybrid BESO with the GPU-accelerated soft-kill BESO using four benchmark problems with more than two million degrees of freedom reveals three key benefits of the proposed hybrid model: reduced execution time, decreased memory consumption, and improved FEA convergence, all of which mitigate the major computational issues associated with BESO.

Topology optimization for maximizing buckling strength using a linear material model

Buckling resistance has gained significant attention in topology optimization due to its profound implications for structural designs. Despite considerable research on buckling-constrained topology optimization, maximizing the critical buckling load factor (BLF) still remains a challenging topic. In this study, an innovative algorithm that utilizes a linear material interpolation scheme is introduced to maximize the buckling resistance of structures. The linear material model offers several advantages, such as obviating the need to select the penalization schemes and penalty values, facilitating straightforward sensitivity analysis, and removing the ambiguous physical meaning of penalization for the stress stiffness matrix. The accuracy of the linear material model for buckling analysis is systematically examined, and the avoidance of stress singularities in low-density regions is investigated. The effectiveness and efficiency of the proposed approach are supported by four buckling optimization design examples, which also demonstrate substantial improvements compared to the existing algorithms.

Topology optimization of porous structures by considering acoustic and mechanical characteristics

Porous structures are widely found in nature and have been extensively studied due to their high stiffness-to-weight ratio, excellent sound absorption, and robustness regarding loading variation. This paper proposes a new topology optimization approach for the design of porous acoustic-mechanical structures based on the three-field floating projection topology optimization (FPTO) method. In order to capture the change of ambiguous structural boundary during topology optimization, the mixed displacement/pressure (u/p) finite element formulation with the linear material interpolation scheme is adopted. The artificial porous features of the structures are generated by imposing the local volume constraint defined by a constant or adaptive filter radius. The 0/1 constraints of the design variables are simulated by the floating projection constraint so that the final design has a clear topology. To verify the effectiveness of the proposed method, some interesting 2D and 3D numerical examples considering their mechanical and acoustic characteristics simultaneously are presented to achieve porous structures by minimizing dynamic compliance, possibly with an additional acoustic constraint. The results also show that the resulting porous features are useful in improving the acoustic performance of the optimized designs.

Microstructure design of porous viscoelastic composites with prescribed relaxation moduli by multi-material topology optimization

This paper presents a multi-material topology optimization method for designing the microstructures of porous viscoelastic composites with prescribed relaxation moduli. It is assumed that the composite materials are microscopically composed of periodic unit cells. The effective relaxation moduli of the composites are obtained using the homogenization method. The density-based topology optimization is utilized, and a related material interpolation scheme is introduced. The design objective is to determine an optimal material distribution that exhibits the prescribed modulus in both the glassy and rubbery regions. The design sensitivity is derived analytically using the adjoint variable method. A filtering scheme is adopted to impose a minimum length scale for each material phase. Several numerical examples are presented to demonstrate the effectiveness of the proposed method. Various microstructures with prescribed relaxation moduli are presented and discussed. In addition, the bounds on the relaxation moduli of the optimized microstructures are compared with theoretical bounds.

A unified material interpolation for topology optimization of multi-materials

Topology optimization is one of the engineering tools for finding efficient design. For the material interpolation scheme, it is usual to employ the SIMP (Solid Isotropic Material with Penalization) or the homogenization based interpolation function for the parameterization of the material properties with respect to the design variables assigned to each finite element. For topology optimization with single material design, i.e., solid or void, the parameterization with 1 for solid and 0 for void becomes relatively straight forward using a polynomial function. For the case of multiple materials, some issues of the equality modeling of each material and the clear 0, 1 result of each element for the topology optimization issues become serious because of the curse of the dimension. To relieve these issues, this research proposes a new mapping based interpolation function for multi-material topology optimization. Unlike the polynomial based interpolation, this new interpolation is formulated by the ratio of the p-norm of the design variables to the 1-norm of the design variable multiplied by the design variable for a specific material. With this alternative mapping based interpolation function, each material are equally modeled and the clear 0, 1 result of each material for the multi-material topology optimization model can be improved. This paper solves several topology optimization problems to prove the validity of the present interpolation function.

On multi-material topology optimisation problems under inhomogeneous Neumann–Dirichlet boundary conditions

This work deals with the topology optimisation of structures made of multiple material phases. The proposed approach is based on non-uniform rational basis spline (NURBS) hyper-surfaces to represent the geometric descriptor related to each material phase composing the continuum, and an improved multiphase material interpolation (MMI) scheme to penalise the stiffness tensor of the structure. In this context, the problem is formulated in the most general case by considering inhomogeneous Neumann–Dirichlet boundary conditions and by highlighting the differences between two different problem formulations. The first one uses the work of applied forces and displacements as cost function and the resulting optimisation problem is not self-adjoint. The second one considers the generalised compliance (related to the total potential energy), and the resulting optimisation problem is self-adjoint. Moreover, the improved MMI scheme proposed here does not require the introduction of artificial filtering techniques to smooth the boundary of the topological descriptors of the material phases composing the structure. The effectiveness of the method is proven on both 2D and 3D problems. Specifically, an extensive campaign of numerical analyses is conducted to investigate the influence of the type of geometric descriptor, of the integer parameters involved in the definition of the NURBS entity, of the type of cost function, of the type of lightness requirement, of the number and type of material phases, of the applied boundary conditions on the optimised topology. • Topology optimisation of multi-material media under mixed boundary conditions • The generalised compliance is used as a measure of the structural stiffness • The work of external actions is not a self-adjoint function (unlike the compliance) • The geometrical descriptor of each material phase is a NURBS hyper-surface • Influence of NURBS parameters and number of materials on the optimal solution

On topology optimization with gradient-enhanced damage: An alternative formulation based on linear physics

Numerous topology optimization formulations have been proposed in order to enhance structural resistance to material failure. A clear line can often be drawn between those methods which attempt to constrain local failure criteria and those that explicitly model the failure physics during the optimization process. In this work the former method is extended in a manner inspired by the mathematical form of typical gradient-enhanced damage models. Importantly, the proposed formulation relies on linear physics during the optimization procedure, which greatly increases its speed and robustness, both of which are essential in industrial applications where large numerical models are typically used. The size effect introduced by using such a numerical model is further investigated and select observations are provided, such as spurious “fin-like” patterns that emerge depending on the type of structure and loading conditions. Finally, the load capacity is verified for each optimized design through a post-optimization verification procedure which is unaffected by the density-based design parameterization and associated material interpolation schemes.

Multi-scale design of multi-material lattice structures through a CAD-compatible topology optimisation algorithm

This work deals with the multi-scale topology optimisation (TO) of multi-material lattice structures. The proposed approach is based on: non-uniform rational basis spline (NURBS) hyper-surfaces to represent the geometric descriptor related to each material phase composing the representative volume element (RVE), an improved multiphase material interpolation (MMI) scheme to penalise the element stiffness tensor of the multi-material RVE, the strain energy-based homogenisation method (SEHM) to carry out the scale transition. In this context, the design requirements are defined at different scales and their gradient is evaluated by exploiting the properties of the NURBS entities and of the SEHM. Moreover, the improved MMI scheme proposed here does not require the introduction of artificial filtering techniques to smooth the topological descriptors of the material phases composing the RVE. The effectiveness of the method is proven on both 2D and 3D problems. Specifically, a sensitivity analysis of the optimised configuration of the RVE to the parameters tuning the shape of the NURBS entity is conducted. Finally, the influence of the starting point and of the macroscopic loads on the optimal solution is investigated.

Three-field floating projection topology optimization of continuum structures

Topology optimization using the variable substitution among three fields can achieve a design with desired solid and/or void features. This paper proposes a three-field floating projection topology optimization (FPTO) method using the linear material interpolation. The implicit floating projection constraint is used as an engine for generating a 0/1 solution at the design field. The substitution filtering and projection schemes enhance the length scale and solid/void features to accelerate the formation of structural topology in the physical field. Meanwhile, the three-field FPTO method can be extended to robust formulation, which obtains the eroded, intermediate, and dilated designs with the same topology. The most distinct feature of the FPTO method lies in the adoption of the linear material interpolation scheme, which makes many topology optimization problems straightforward. As an example, the proposed three-field FPTO algorithm is further applied to the design of shell-infill structures using the linear multi-material interpolation scheme. The distribution of the shell material is generated through a simple filtering scheme, and the shell thickness is accurately controlled by the filter radius. Numerical examples are presented to demonstrate the effectiveness and advantage of the proposed three-field FPTO method.

Inner Sequential Single Solid Method for Layout Optimization of Multi-Materials

In multiple materials layout optimization, the optimal solution is difficult to achieve due to the number of design variables being too large and the material interpolation scheme becoming complicated when using density-like method. To bypass this problem, an inner sequential single solid optimization (ISSSO) method is presented in this study. There are two types of iterations in the optimization process of this method: the inner and outer iterations. In the inner iteration, the original layout optimization of many materials is replaced with a series of single solid sub-optimizations. In each sub-optimization, only one solid is chosen to update and the rest keeps unchanged. The inner iteration stops when all the solids are updated consequently. The next loop of inner iteration starts when the current solution does not meet the requirement of convergence. The outer iteration starts only if the current inner iteration comes to an end. In the outer iteration, the new inner iterations for update of all the solid materials are carried out. The effectiveness of ISSSO method is verified by comparing the calculation results with other methods.

Topology optimization of acoustic systems with a multiconstrained BESO approach

This work details a new acoustic topology optimization methodology with applications on the design of systems composed of rigid and porous materials. The Bi-directional Evolutionary Structural Optimization (BESO) algorithm is combined with the Virtual Temperature Method to minimize the occurrence of air cavities and rough surfaces inside rigid and porous domains, hence configuring a multiconstrained optimization approach. The modeling of porous materials is done by the Johnson–Champoux–Allard (JCA) formulation, while the Finite Element Method is used to approximate the governing equations of the physical system. Two different optimization problems are considered separately: first, a rigid–acoustic metasurface is optimized to reduce regional sound pressure levels (SPL) in a set of observed frequencies, while also considering wind permeability through the structure. Secondly, a coupled poro-acoustic absorptive system is treated in order to enhance the sound absorption coefficient in the low frequency range. Both problems are systematized by the implementation of acoustic–rigid and acoustic–porous material interpolation schemes, respectively. The effectiveness of the proposed approach is explored through numerical examples. Here, it is remarked that the methodology maintains the uniformity of rigid barriers, by guaranteeing the absence of internal holes to them. In addition, well-defined cavities are formed in porous domains, increasing their absorption coefficients, but without inflicting macroscopic closed spaces within such structures. In these cases, comparison with appropriate literature is also provided. • Multiconstrained topology optimization algorithm for acoustic problems. • Implementation of the Virtual Temperature Method for connectivity. • Optimization of metasurfaces for reduction of sound pressure levels. • Optimization of porous structures for enhancements of sound absorption coefficient.

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.

A novel discrete–continuous material orientation optimization model for stiffness-based concurrent design of fiber composite

The cooperation of tailoring fiber angles and designing its macro layout is an excellent approach to improve the performance of composite structures and achieve lighter weight. This article proposes a novel discrete–continuous optimization model for the concurrent optimization of the macro structural topology and local fiber angle for the fiber-reinforced composite structure. Therein, the novel modeling consists of a two-step strategy, where the normal distribution function is adopted to form the discrete material interpolation scheme firstly and then the continuous angle design on fine angle subintervals is conducted. Because it requires the fixed discrete design variables regardless of the number of angle subintervals and simultaneously achieves the continuous fiber angle design, it can save computational cost and provide an infinite design domain. Furthermore, the subdividing angle interval reduces the risks of falling into the local optimum solution for continuous angle design. Besides, the fiber discontinuity may cause stress concentration and manufacturing problems in the intersection of discontinuous fiber paths. Therefore, a linear density technique is used to achieve fiber continuity. Finally, a concurrent topology optimization model is built based on the proposed novel model. And several numerical examples are well-organized and presented to illustrate the validity of the proposed model.