Solid Isotropic Material With Penalization Research Articles

Transient heat conduction in multi-material topology optimization of thermoelastic structures involving dynamic constraints

This work presents a modified solid isotropic material with penalization (SIMP) for conducting dynamic-constrained multi-material topology optimization (MMTO) of thermoelastic structures subjected to transient thermal conduction. The novelty of this study is the multi-material optimization of instantaneous temperature conduction under specific frequency constraints, which has never been studied before. The proposed method has been developed for general models of multi-material interpolations such as thermal stiffness, heat capacity, and thermal stress coefficient. The temperature environment is formulated by applying prescribed mechanical and thermal loads that are exposed over a period of time. In this work, we focus on the objective of minimizing compliance in a transient heat conduction structure while simultaneously imposing constraints on the dynamics. With high temperatures, the stiffness of the optimization results decreases while stability is still guaranteed. The sensitivity of the objective function is subsequently calculated using the discretize-then-differentiate approach for the dynamic problem as well as the adjoint variable method. This study examines the impact of thermal fields and different time increments in conjunction with dynamic criteria. The outcomes of heat conduction result in notable alterations under identical dynamic constraint levels.

Equivariant neural operators for gradient-consistent topology optimization

Abstract Most traditional methods for solving partial differential equations (PDEs) require the costly solving of large linear systems. Neural operators (NOs) offer remarkable speed-ups over classical numerical PDE solvers. Here, we conduct the first exploration and comparison of NOs for three-dimensional topology optimization. Specifically, we propose replacing the PDE solver within the popular Solid Isotropic Material with Penalization (SIMP) algorithm, which is its main computational bottleneck. For this, the NO not only needs to solve the PDE with sufficient accuracy but also has the additional challenge of providing accurate gradients which are necessary for SIMP’s density updates. To realize this, we do three things: (i) We introduce a novel loss term to promote gradient-consistency. (ii) We guarantee equivariance in our NOs to increase the physical correctness of predictions. (iii) We introduce a novel NO architecture called U-Net Fourier neural operator (U-Net FNO), which combines the multi-resolution properties of U-Nets with the Fourier neural operator (FNO)’s focus on local features in frequency space. In our experiments we demonstrate that the inclusion of the novel gradient loss term is necessary to obtain good results. Furthermore, enforcing group equivariance greatly improves the quality of predictions, especially on small training datasets. Finally, we show that in our experiments the U-Net FNO outperforms both a standard U-Net, as well as other FNO methods.

Multi-scale concurrent topology optimization of cellular structures with multiple microstructures for minimizing dynamic response in the time domain

This paper proposes a concurrent topology optimization method for optimizing the structures that are periodically filled with multiple microstructures excited by dynamic response in the time domain. At the macroscale, different microstructures are considered as different materials. To generate the various distribution of different microstructures, a multi-material interpolation approach based on the Solid Isotropic Material with Penalization (SIMP) is integrated. At the microscale, the energy-based homogenization method (EBHM) is employed to determine the macroscopic effective properties of the microstructures. All macroscale elements with identical material are represented by a distinct microstructure. For the proportional damping model, the HHT-α is implemented as a time integration technique to obtain the dynamic response of multi-scale and assumed multi-material structures. The objective function of the topology optimization problem is to minimize the mean dynamic compliance. Combined differentiate-then-discretize method with the adjoint variable method, the sensitivity analysis applies the gradient-based Zhang-Paulino-Ramos Jr. (ZPR) algorithm or Method of Moving Asymptotes (MMA) to update the design variables under multiple constraints in the time and space-discretized system. The proposed approach is numerically performed through 2D and 3D examples to demonstrate its effectiveness.

Experiments and investigation of planar high-strength steel joints with Additive Manufacturing

This paper investigates the application of high-strength steel in planar steel joints, specifically X-joints and T-joints, which are crucial connecting components in structural engineering, ensuring structural stability in buildings, bridges, and tower cranes. Conventional steel joints, mainly constructed from low-carbon steel, encounter challenges arising from the low ductility of high-strength steel and its strict welding criteria. The Solid Isotropic Material with Penalization (SIMP) method is employed to perform topology optimization on X-joints with different brace width to chord width ratios and T-joints, enabling the manufacturing of high-strength steel components through the Additive Manufacturing method (AM). The impact of different additive manufacturing orientations on material strength is tested, with results indicating improved mechanical properties when the loading direction is perpendicular to the normal direction of the additive manufacturing layer. Compression tests are conducted using 2D/3D Digital Image Correlation methods (2D/3D-DIC) alongside traditional measurement techniques. The results indicate that the stress distribution of the optimized joint is more rational, and the strength of the optimized joint has improved compared to the tubular joint. The optimized joint exhibits a more distinct load path and apparent failure mode. This optimized joint enables a uniform stress redistribution, enhancing ductility and achieving a strong-joint and weak-component structural mechanism that satisfies the serviceability limit state (SLS). In comparison to non-penetrative tubular joints, no significant cracks are observed under identical loading conditions.

Design of metadevices based on isotropic materials for eigenfrequency recovery in lightened structures

This work presents a novel approach for the design of metadevices conceived to recover a prescribed set of the natural eigenfrequencies of a structure lightened by machining arbitrary shape holes, using classical topology optimization (TO) techniques. This TO metadevice design (TOMD) is performed by minimizing the error of an eigenfrequency recovery task, using the finite element method (FEM) as an analysis tool, ensuring manufacturability via the solid isotropic material with penalization (SIMP) approach with Heaviside filtering to reduce intermediate-density areas. The spatial variation of mechanical properties derives from the macroscopically distinguishable dual material distribution obtained from the TOMD problem. The methodology was first tested on lightened thin steel plates and then extended to a real-world 3D mechanical component with additional mechanical performance testing. In all cases, the optimized metadevice yielded lower error in recovering eigenfrequencies compared to the lightened component without the metadevice. Furthermore, the metadevice generally achieved further mass reduction, demonstrating its potential for lightweight design while restoring its original eigenfrequencies.

Multi-scale design of composite material structures for maximizing fundamental natural frequency

This study deals with the topology optimization (TO) of linear elastic composite structures considering fundamental natural frequency maximization. The concept of functionally graded structures (FGS) is used for anisotropic composite material structure design to improve the structural dynamic performance. The rational approximation of material properties (RAMP) method was employed as an alternative to the solid isotropic material with penalization (SIMP) method to improve the convergence stability for eigen-frequency problem. In the SIMP method, the occurrence of local eigen-modes at low frequencies in the void region causes ill-convergence, whereas the RAMP method prevents local eigen-modes from occurring. The method of moving asymptotes is used as an optimizer. Numerical verification was performed for three structural types: cantilever beam, double-clamped beam, and four-pinned beam structure. In addition, for cantilever beam design, bi-objective optimization for maximizing fundamental natural frequency as well as minimizing structural compliance was performed using the adaptive weighting method that quantitatively reflects the designer's preference by adjusting the weight for each objective function at each iteration.

Multi-objective optimization design of power pack structures based on floating projection topology optimization

Battery packs are vital components for storing and releasing energy. However, during driving, the intensity of heat emission and mechanical performance of electric vehicles are interrelated but conflicting indicators. To address this problem, this study topologically optimizes the power pack support structure with multiple objectives of minimizing compliance and heat dissipation, aiming to achieve a comprehensive and integrated design scheme. A floating projection topology optimization (FPTO) method is proposed, which normalizes the objectives of structural compliance and heat dissipation and derives the sensitivity. FPTO demonstrates superior objective function values and smoother topology configuration compared to the solid isotropic material with penalization (SIMP) method. Finally, the proposed method is applied to the power pack support structure. The results demonstrate that the method obtains a design solution with excellent mechanical performance and high heat dissipation performance with a suitable trade-off. The lightweight design of the overall structure of the battery pack is achieved.

Shell-infill composite structure design based on a hybrid explicit- implicit topology optimization method

The present paper introduces a hybrid explicit–implicit topology optimization method for shell-infill composite structure design. The proposed approach effectively combines the advantages of the explicit Moving Morphable Component (MMC) method, which describes structural topology only using a set of geometric parameters, and the implicit Solid Isotropic Material with Penalization (SIMP) method, which offers greater design freedom for characterizing the structural features. Compared to the existing methods for shell-infill structure design, the proposed approach can obtain optimized shell-infill structures with complex infill topology without resorting to complex filtering/projection operations. Numerical examples demonstrate the effectiveness of the proposed approach.

ResUNet involved generative adversarial network-based topology optimization for design of 2D microstructure with extreme material properties

Topology optimization is one of the most common methods for design of material distribution in mechanical metamaterials, but resulting in expensive computational cost due to iterative simulation of finite element method. In this work, a novel deep learning-based topology optimization method is proposed to design mechanical microstructure efficiently for metamaterials with extreme material properties, such as maximum bulk modulus, maximum shear modulus, or negative Poisson’s ratio. Large numbers of microstructures with various configurations are first simulated by modified solid isotropic material with penalization (SIMP), to construct the microstructure data set. Subsequently, the ResUNet involved generative and adversarial network (ResUNet-GAN) is developed for high-dimensional mapping between optimization parameters and corresponding microstructures to improve the design accuracy of ResUNet. By given optimization parameters, the well-trained ResUNet-GAN is successfully applied to the microstructure design of metamaterials with different optimization objectives under proper configurations. According to the simulation results, the proposed ResUNet-GAN-based topology optimization not only significantly reduces the computational duration for the optimization process, but also improves the structure precise and mechanical performance.

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.

Investigating best algorithms for structural topology optimization

This study investigates the topology optimization problem using various optimization approaches, taking inspiration from the 99-line MATLAB code developed by Sigmund. The educational MATLAB code is based on the Solid Isotropic Material with Penalization (SIMP) model of the artificial material density method. The objective is to minimize the compliance function with a weight constraint, with the design variables being the densities of all elements. The aim is to identify a more efficient optimization technique as an alternative to the commonly used optimality criteria algorithm provided by other MATLAB built-in tools. Two types of optimization algorithms are examined: gradient-based methods such as Interior-Point, Sequential Quadratic Programming (SQP), and Active-Set, as well as metaheuristic methods including the Genetic Algorithm. The results are verified and validated by comparing them with existing literature, demonstrating good agreement. Performance assessments are conducted to compare the results obtained from these algorithms in terms of quality and computational efficiency. The numerical findings indicate that the interior-point method outperforms the other investigated methods, although the optimality criteria algorithm remains the most efficient for solving topology optimization problems.

Topology optimization of Shell–Infill structures with enhanced edge-detection and coating thickness control

Topology optimization is a valuable tool in additive manufacturing, but its application requires careful consideration of various constraints to refine the optimization outcome. In additive manufacturing, components often consist of two distinct regions: a solid shell forming the exterior and a porous infill occupying the interior. This work introduces a novel approach based on the SIMP (Solid Isotropic Material with Penalization) method, aiming to optimize both the shell and infill concurrently. The proposed method incorporates a series of filtering processes that effectively separate the structure into two phases: the shell and the infill. This separation allows for precise control over the desired coating thickness by employing the SUSAN edge-detector method and introducing a new parameter designed for explicit thickness control. Additionally, a new constraint based on the structural skeleton has been introduced to ensure the preservation of a minimum length scale.

Design of the multiphase material structures with mass, stiffness, stress, and dynamic criteria via a modified ordered SIMP topology optimization

This paper, a multi-material topology optimization method (MMTO) is presented for two-dimensional structures using modified ordered solid isotropic material with penalization (SIMP) under multiple constraints such as mass, stiffness, frequency, and stress. The element density is obtained by a density filtering and the Heaviside approximation, which are combined to avoid gray elements and numerical issues of stress. Effective improvement of multiphase material topology optimization was performed including the criteria for stiffness, stress, and free vibration. P-norm stress aggregation is employed to calculate the maximum von Mises stress and sensitivity of von Mises stress is formulated by the adjoint method. The natural frequency and stress constraints are solved simultaneously to enhance the stability and reduce the stress concentration of the optimized structures, which are not much covered in the literature. The multi-mode phenomenon in the eigenfrequency problem is treated by the simply proposed technique. Results indicate that the multi-material designs with multi-constraints can be illustrated and optimized effectively. The optimized topologies, for stress minimization, indicate that the maximum stress can be significantly reduced compared with compliance minimization. The maximum von Mises stress of the topological results considering the maximum stiffness with natural frequency and mass constraints is higher than that maximum stiffness with only mass constraints, which indicates that the optimized structure created considering the eigenvalues is safer. The proposed approach can obtain a reasonable result that effectively controls the stress level and reduces the stress concentration at the high stress regions of multi-phase material structures with multi-constraints. Benchmark numerical examples are presented to confirm the effectiveness of the presented method.

Towards the saving potentials using a hybrid topology optimization: Application of a coupled deterministic and empirical approach solving a geotechnical optimization problem

AbstractThis article presents the application of the TOSS (topology optimization for stiffness and strength) algorithm, a hybrid optimization strategy for topology optimization, used for the design of a strip foundation as a basic geotechnical structure. One of the challenges of this field is to improve structural performance by using less material than conventional structures. Therefore, the SIMP (solid isotropic material with penalization) is applied to optimize the foundation concerning its deformational behavior. The downstream SKO (soft kill option) then solves for optimality concerning the foundation's stress distribution. Numerical experiments with the TOSS algorithm reveal great potential for material savings. The coupled solution presents the benefit of reducing emissions due to material savings of more than 20% in comparison to the individual SIMP and SKO algorithms. A positive by‐product of the method is that stress peaks under the foundation are reduced due to the homogenization of stresses as a result of SKO optimization, thus creating the conditions for better durability of the foundation.

Topology optimization of extruded beams modeled with the XFEM for maximizing their natural frequencies

In this paper, an efficient topology optimization approach is developed for maximizing the fundamental natural frequency of extruded beams. Mass fraction and static compliance bounds are defined using inequality-type constraints in the optimization problem. An XFEM approach, previously proposed by the authors for analyzing beam elements, is extended herein to compute the natural frequencies of the beam. The method allows for 3D modeling of beams with a significant reduction in the number of degrees of freedom and therefore also yields efficient optimization procedure. This reduction is made possible by incorporating global enrichment functions in the longitudinal direction, which enables a significant reduction in the number of elements in that direction without loss of accuracy. A nonlinear optimization problem is formulated using continuous density-based design variables that represent the material distribution in the beam’s cross-section. The optimization problem is then solved using a gradient-based approach with analytical sensitivities. The well-known Solid Isotropic Material with Penalization (SIMP) method is used to acquire discrete solutions. We study the optimal design of short and long beams. It is shown that for short beams, localized vibration modes appear within the cross-section, leading to a significant distortion deformation mode of the cross-section. The optimized design of the long beam shows global deformation modes with an increase of 15% in the fundamental frequency compared with a non-optimized design consisting of a hollow rectangular cross-section with the same mass.

Evaluation of Neural Network-based Derivatives for Topology Optimization

Abstract Neural networks have gained popularity for modeling complex non-linear relationships. Their computational efficiency has led to their growing adoption in optimization methods, including topology optimization. Recently, there have been several contributions towards improving derivatives of neural network outputs, which can improve their use in gradient-based optimization. However, a comparative study has yet to be conducted on the different derivative methods for the sensitivity of the input features on the neural network outputs. This paper aims to evaluate four derivative methods: analytical neural network's Jacobian, central finite difference method, complex step method, and automatic differentiation. These methods are implemented into density-based and homogenization-based topology optimization using multilayer perceptrons (MLPs). For density-based topology optimization, the MLP approximates Young's modulus for the solid-isotropic-material-with-penalization (SIMP) model. For homogenization-based topology optimization, the MLP approximates the homogenized stiffness tensor of a representative volume element, e.g., square cell microstructure with a rectangular hole. The comparative study is performed by solving two-dimensional topology optimization problems using the sensitivity coefficients from each derivative method. Evaluation includes initial sensitivity coefficients, convergence plots, and the final topologies, compliance, and design variables. The findings demonstrate that neural network-based sensitivity coefficients are sufficient for density-based and homogenization-based topology optimization. The neural network's Jacobian, complex step method, and automatic differentiation produced identical sensitivity coefficients to working precision. The study's open-source code is provided through an included Python repository.

Bisection Constraint Method for Multiple-Loading Conditions in Structural Topology Optimization

Topology optimization (TO) is currently a focal point for researchers in the field of structural optimization, with most studies concentrating on single-loading conditions. However, real engineering structures often have to work under various loading conditions. Approaches addressing multiple-loading conditions often necessitate subjective input in order to determine the importance of each loading condition, aiming for a compromise between them. This paper proposes a so-called bisection constraint method (BCM), offering a unique, user-preference-independent solution for TO problems amidst multiple-loading conditions. It is well-known that minimizing the system’s compliance is commonly used in TO as the objective. Generally, compliance is not as sufficient as stress to be used as a response to evaluate the performance of structures. However, formulations focusing on minimizing stress levels usually pose significant difficulties and instabilities. On the other hand, the compliance approach is generally simpler and more capable of providing relatively sturdy designs. Hence, the formulation of min–max compliance is used as the target problem formulation of the proposed method. This method attempts to minimize compliance under only one loading condition while compliances under the remaining loading conditions are constrained. During the optimization process, the optimization problem is automatically reformulated with a new objective function and a new set of constraint functions. The role of compliance under different loading conditions, i.e., whether it is to be treated as an objective or constraint function, might be changed throughout the optimization process until convergence. Several examples based on the solid isotropic material with penalization (SIMP) approach were conducted to illustrate the validity of the proposed method. Furthermore, the general effectiveness of the compliance approach in terms of stress levels is also discussed. The calculation results demonstrated that while the compliance approach is effective in several cases, it proves ineffective in certain scenarios.

Integrated optimization of components’ layout and structural topology with considering the interface stress constraint

Structural designs in the engineering field commonly require the prior embedding of several fixed-shape components to satisfy desired performance or functional requirements. In the past few decades, topology optimization has been regarded as an effective approach to deal with the optimization problems of continuous structures embedded with multiple components. However, the interfaces between the host supporting structure and embedded components are assumed perfect-bonded, which may result that the optimal design occurs interface failure and structural damage under extreme or long-time service conditions. In this study, a new topology optimization method of structures with embedded components considering the tension/compression-asymmetric interface stress constraint is proposed under the solid isotropic material with penalization (SIMP) method framework. This proposed method optimizes simultaneously the topology of the host structure and the layout of embedded components. A new interpolation model is developed to determine interface layers between the host structure and embedded components. Then, interface stress constraints are derived and imposed on each element of interface layers. For relieving the computational burden caused by enormous stress constraints, the global p-norm stress aggregation method is adopted to quantify the peak stresses. Subsequently, the interface stress constraint is introduced into the integrated optimization of multi-component layout and structural topology, the objective function is structural compliance and constraint functions are composed of volume, non-overlapping and tension/compression-asymmetric interface stress constraints. The corresponding sensitivity analyses are computed by using the adjoint method. Several two-dimensional and three-dimensional numerical examples are provided to verify the effectiveness of the proposed method.

Multi–materials topology optimization using deep neural network for coupled thermo–mechanical problems

In this paper, a density-based topology optimization method using neural networks is used to design domains composed of multi-materials under combined thermo–mechanical loading. The neural network produces the topology density field by minimizing a loss function defined for the problem. Length scale control is conducted using a Fourier space projection. JAX, a high-performance automatic differentiation (AD) Python library is used to build an end-to-end differentiable neural network. The model can handle the sensitivity analysis automatically using the backpropagation process (automatic differentiation of loss function) and the need for solving adjoint equations manual sensitivity analysis is removed. The optimization problem is solved by minimizing the loss function of the network and other optimization algorithms such as the Method of Moving Asymptotes (MMA) are not required. The performance of the method is demonstrated through several examples and compared with those obtained from the popular Solid Isotropic Material with Penalization (SIMP) method. The network is able to handle high-resolution re-sampling, resulting in a more refined and smooth variation of optimal topologies.

Development of reduced volume endosseous cuspid tooth implant using topology optimization by SIMP technique for improved osseointegration

The article aims to design and develop a topology-optimized endosseous cuspid tooth implant of the maxilla region. The manuscript presents a numerical analysis of the resulting von Mises stresses and effective strain resulting in the topology-optimized implant with occlusal loading of 110 N. Solid Isotropic Material with Penalization (SIMP) method is employed for topology optimization and four different models, namely model-1, model-2, model-3, and model-4, are developed based on volume reduction rates of 8%, 16%, 24%, and 32%, respectively. FEA results highlight that the maximum stress and strain in the screw increases with volume reduction rates. The comparative analyses of the resulting stresses in the compact and cancellous bone along with the strain in the screw led to the conclusion that model-1, model-2, and model-3 resulted in moderate stresses on compact and cancellous bone compared to the original model of the implant. However, the screw and bones are subjected to maximum stress and strain in the model-4. The study concludes that model-2, with 16% reduced volume and 14.2% reduced mass as compared to the original implant, may be considered as the optimized design of the model. The resulting model offers a significant reduction in the weight and volume with a minor increase in effective stress and strain without negatively impacting the functionality and bio-mechanical performance of the implant. The optimized dental implant prototype is also fabricated as a proof of concept by the Fused Deposition Modelling process.