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.

Computational design of auxetic microstructures via stress-based topology optimization

Material design plays a pivotal role in industries dedicated to lightweight manufacturing. In response to the needs of such industries, topology optimization (TO) has evolved beyond structural optimization to include designing material microstructures for improved structural performance. While conventional approaches often prioritize stiffness-oriented designs as a foundational aspect of structural integrity, it is equally imperative to emphasize strength-based design principles to ensure robustness against localized stress concentrations and potential failure modes. This paper explores the application of stress-constrained topology optimization (SCTO) for designing auxetic microstructures. The study commences with a comprehensive parametric analysis to examine the influence of the P-norm parameter, variations in volume fraction, and initial design adjustments on the resulting microstructures. Several optimized structures, accompanied by stress contours, are generated to showcase how the optimization process responds to these parameters. The approach employs the density-based solid isotropic material with penalization material interpolation scheme, and it utilizes the stress penalization technique to mitigate the singularity phenomenon. The multi-constraint optimization employs the method of moving asymptotes algorithm to update design variables. The effectiveness of the proposed approach to utilize actual strain fields, derived from macroscopic finite element analysis under service loads, for TO of an auxetic microstructure according to strength requirements is demonstrated. The application of the designed microstructure is explored through its integration into an auxetic cushion pad. This analysis confirms the microstructure’s effectiveness in practical applications, demonstrating its potential for broader usage in similar applications.

Evolutionary topology optimization of fiber reinforced composite laminates for maximum stiffness

In this study, a topology optimization technique based on the bi-directional evolutionary structural optimization (BESO) method is developed to maximize the stiffness of fiber reinforced composite laminates. The elastic properties of single-layer and multi-layer composite laminates are characterized and a composite material interpolation scheme is introduced. The effectiveness of the developed BESO method for the single-layer and multi-layer composite laminates are verified by three classical examples, namely a cantilever, a Michell-type structure, and an L-bracket. Then, based on the analysis of the topological results, a selection suggestion and verification criteria for layers with appropriately matched fiber angles are proposed and verified by two additional examples. This study proves that the BESO method is an effective technique for the topology optimization of composite laminates for maximum stiffness, and the proposed suggestion and criteria are useful for avoiding large computational costs and empirical selection of layers.

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.