Solid Isotropic Material With Penalization Method Research Articles

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.

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.

ТОПОЛОГИЧЕСКАЯ ОПТИМИЗАЦИЯ ЭЛЕМЕНТОВ КОНСТРУКЦИИ РОБОТИЗИРОВАННОЙ ЯЧЕЙКИ

The presented article describes the technology of topological optimization. The main methods of topological optimization are also described: ESO/BESO; SIMP-method; ESO-Simp-method; Level-Set-method. Each of the methods has its advantages, but they have common disadvantages, namely: the problem of the "chessboard" and the dependence on grid partitioning. Topological optimization allows you to reduce the weight of the product, while maintaining the strength characteristics. Based on the product obtained after optimization, its production technologies were considered. These include traditional technologies (casting, stamping, milling, etc.) and additive manufacturing technologies. Software with topological optimization functionality was presented and described. These are mainly commercial CAD/CAE systems (OptiStruct, Simulia Tosca, ANSYS and MSC Nastran). The main industries that use the method of topological optimization are the aerospace industry, mechanical engineering, medicine, robotics and others. In this paper, we consider the process of reducing the weight of a robotic cell trolley using the SIMP (Solid Isotropic Material with Penalization) method, on which a robotic arm is attached. The functionality of the CAD/CAM/CAE-system NX from Siemens PLM Software was used for the topological optimization of the robotic cell trolley. Generalizing conclusions are made that topological optimization makes it possible to obtain geometric shapes of a product with a minimum mass while maintaining strength and rigidity.

Τopology Optimization under a Single Displacement Constraint Using a Strain Energy Criterion

Based on a previous concept that has been successfully applied to the sizing optimization of truss and frame structures, this work extends and improves the strain energy criterion in the topology optimization of 2D continuum structures under a single displacement constraint. To make the proposed methodology transparent to other researchers and at the same time meaningful, the numerical value of the displacement constraint was taken to be equal to that obtained through the well-known Solid Isotropic Material with Penalization (SIMP) method under the same boundary conditions and the same external forces. The proposed method is more efficient than the SIMP method while leading to topologies very close to those obtained by the latter.

Energy absorption and vibration mitigation performances of novel 2D auxetic metamaterials

This paper proposes novel unit cells that can be arranged periodically constructing disparate auxetic metamaterials with negative Poisson’s ratio for engineering applications. The two-dimensional auxetic unit cells are designed utilizing the mathematical model of the modified Solid Isotropic Material with Penalization (SIMP) topological optimization method with the objective of maximizing the magnitude of the negative Poisson’s ratio. The deformation mechanisms of the auxetic structures are controlled effectively by straight and oblique struts attached at rotation joints since the optimum designs are a combination of chiral and re-entrant unit cells. Three structures with optimum auxetic designs are chosen to verify the modified SIMP method using numerical simulation. Additionally, the energy absorption and vibration mitigation of auxetic metamaterials have been extensively studied. The results showed a significant impact of negative Poisson’s ratio on the considered dynamic properties of the auxetic structures.

Topology optimization of heat transfer and elastic problems based on element differential method

The element differential method (EDM) is a robust and efficient strong-form numerical method, which has attracted a lot of attention of the researchers about the numerical methods since proposed. EDM has higher stability than some other strong-form methods and is more efficient than conventional Galerkin FEM. In this paper, the EDM is combined with Solid Isotropic Material with Penalization (SIMP) method to solve 2–3D topological optimization problems under different conditions. The key of the paper is to derive the objective function of second-order EDM element, which is suitable for the SIMP-based topology optimization process. And the sensitivity is solved by Optimality Criteria (OC) method. In the article, topology optimization examples are considered in different mechanics or thermal loads. In the topology optimization of the elastic problem of 3D cantilever beam, the EDM-based SIMP method can reduce the structural compliance by 17∼27% compared with SIMP method based on FEM. And in the heat transfer problems of the 3D plate, the EDM-based SIMP method reduced the compliance by 3.4%. The results show that the present method has good accuracy, efficiency and robustness in topology optimization of 2D and 3D minimum compatibility problems.

Topology optimization using super-resolution image reconstruction methods

This paper proposes a new topology optimization method to obtain super-resolution images without increasing mesh refinement by using various methods. For traditional process, low-resolution (LR) images are fed into the Solid Isotropic Material with Penalization (SIMP) and Optimality Criteria (OC) methods. Here, the trained super-resolution images are added to the inner loops to reconstruct the topology and used to obtain high-resolution (HR) images from the LR images at the end of each iteration. After finishing the reconstruction process, the main topology optimization method recovers the original size images from the HR images for the next iteration. Several examples are presented to demonstrate the effectiveness of the proposed method. The final topologies provide noticeably improvement over those of typical SIMP method and create a much sharper and higher contrast images. Moreover, the proposed strategy using the super-resolution image reconstruction methods can give valuable innovation for conventional topology optimization process.

Topological design of the hybrid structure with high damping and strength efficiency for additive manufacturing

PurposeThree-dimensional (3D) printing provides more possibilities for composite manufacturing. Composites can no longer just be layered or disorderly mixed as before. This paper aims to introduce a new algorithm for dual-material 3D printing design.Design/methodology/approachA novel topology design method: solid isotropic material with penalization (SIMP) for hybrid lattice structure is introduced in this paper. This algorithm extends the traditional SIMP topology optimization, transforming the original 0–1 optimization into A–B optimization. It can be used to optimize the spatial distribution of bi-material composite structures.FindingsA novel hybrid structure with high damping and strength efficiency is studied as an example in this work. By using the topology method, a hybrid Kagome structure is designed. The 3D Kagome truss with face sheet was manufactured by selective laser melting technology, and the thermosetting polyurethane was chosen as filling material. The introduced SIMP method for hybrid lattice structures can be considered an effective way to improve lattice structures’ stiffness and vibration characteristics.Originality/valueThe fabricated hybrid lattice has good stiffness and damping characteristics and can be applied to aerospace components.

Isogeometric topology optimization of strain gradient materials

In this paper optimal topologies of isotropic linear elastic strain gradient materials are investigated by means of isogeometric topology optimization. The employment of strain gradient theory allows not only to capture the microstructural effects of materials but also to regularize stress/strain concentration phenomena and to address the so-called wedge forces. Isogeometric analysis is used in order to meet the requirements of higher-order continuity of strain gradient elasticity theory. For the purpose of determining the constitutive parameters of strain gradient materials, a novel experiment is conceived by using the Digital Image Correlation (DIC) technique. The so-called SIMP (Solid Isotropic Material with Penalization) method is applied to interpolate the material stiffness tensors (including the fourth-order and the sixth-order stiffness tensor) by power law with penalty. Benchmark numerical experiments are conducted to illustrate the computational effectiveness and numerical robustness of the proposed method.

An improved ordered SIMP approach for multiscale concurrent topology optimization with multiple microstructures

Cellular structures are often used to improve the stiffness, fatigue strength, damage tolerance, and other superior functioning properties of structures in engineering. This paper presents a novel multiscale concurrent topology optimization method to optimize structures which are periodically filled with multiple microstructures and connected by solid interfaces. This method generates better structural performance and does not need to preprocess the initial design domain. It enables to save the computational resources, and improves the freedom of structural optimization design to a certain extent. Solid interface layers are built to connect different microstructure blocks, and hence the full degrees of design freedom of microstructures can be further explored. At the macroscale, a novel piecewise projection and a series of gradient-based filtering operations are proposed to distinguish the microstructure blocks and interface layers, respectively. Then, an improved ordered solid isotropic material with penalization (SIMP) method is proposed to optimize the spatial distribution of different microstructures with an affordable computational cost. At the microscale, the microstructures are generated by the numerical homogenization method. Microstructures with different volume constraints are treated as different materials, and the volume fraction limit value of the microstructure corresponds to the design variable of the improved ordered SIMP method. Finally, the compliance minimization problem under the constraint of material volume fractions is investigated, and sensitivity analysis is derived. Several 2D and 3D numerical examples are provided to demonstrate the effectiveness of the proposed method.

Infill topology and shape optimization of lattice‐skin structures

AbstractLattice‐skin structures composed of a thin‐shell skin and a lattice infill are widespread in nature and large‐scale engineering due to their efficiency and exceptional mechanical properties. Recent advances in additive manufacturing, or 3D printing, make it possible to create lattice‐skin structures of almost any size with arbitrary shape and geometric complexity. We propose a novel gradient‐based approach to optimizing both the shape and infill of lattice‐skin structures to improve their efficiency further. The respective gradients are computed by fully considering the lattice‐skin coupling while the lattice topology and shape optimization problems are solved in a sequential manner. The shell is modeled as a Kirchhoff–Love shell and analyzed using isogeometric subdivision surfaces, whereas the lattice is modeled as a pin‐jointed truss. The lattice consists of many cells, possibly of different sizes, with each containing a small number of struts. We propose a penalization approach akin to the SIMP (solid isotropic material with penalization) method for topology optimization of the lattice. Furthermore, a corresponding sensitivity filter and a lattice extraction technique are introduced to ensure the stability of the optimization process and to eliminate scattered struts of small cross‐sectional areas. The developed topology optimization technique is suitable for nonperiodic, nonuniform lattices. For shape optimization of both the shell and the lattice, the geometry of the lattice‐skin structure is parameterized using the free‐form deformation technique. The topology and shape optimization problems are solved in an iterative, sequential manner. The effectiveness of the proposed approach and the influence of different algorithmic parameters are demonstrated with several numerical examples.

Investigation on numerical analysis and mechanics experiments for topology optimization of functionally graded lattice structure

A physical driven method for constructing functionally graded lattice structure is proposed by extending the conventional Solid Isotropic Material with Penalization (SIMP) method with linear buckling load factors and total volume constraints rather than introducing local volume constraints. It can efficiently and effectively construct functionally graded lattice structure by using the SIMP material model for topology optimization with different buckling load factors according to the linearized buckling criteria and the same volume fraction, which greatly improved both the stability and stiffness of the optimal structure. Furthermore, the optimized structures of the illustrative cases are manufactured by means of the fused deposition modeling. Both the numerical analysis and mechanics experiments are further conducted to illustrate the effectiveness of the proposed method compared with the structural topology optimization with SIMP method and the infill structure. To the best of our knowledge, it is the first time to validate the topology optimization of functionally graded lattice structure via both numerical analysis and mechanics experiments.

A Novel Method for Structural Lightweight Design with Topology Optimization

Topological optimization is an innovative method to realize the lightweight design. This paper proposes a hybrid topology optimization method that combines the SIMP (solid isotropic material with penalization) method and genetic algorithm (GA), called the SIMP-GA method. In the method, SIMP is used to update the chromosomes, which can accelerate convergence. The filtering scheme in the SIMP method can filter unconnected elements to ensure the connectivity of the structure. We studied the influence of varying the filtering radius on the optimized structure. Simultaneously, in the SIMP-GA method, each element is regarded as a gene, which controls the population number to a certain extent, reduces the amount of calculation, and improves the calculation efficiency. The calculation of some typical examples proves that the SIMP-GA method can obtain a better solution than the gradient-based method. Compared with the conventional genetic algorithm and GA-BESO (Bi-directional Evolutionary Structural Optimization) method, the calculation efficiency of the proposed method is higher and similar results are obtained. The innovative topology optimization method could be an effective way for structural lightweight design.

Reducing Weight of Freight Bogie Bolster Using Topology Optimization

Nowadays, the weight minimization of a structure without affecting the initial performance is a challenging issue for design engineers. Weight reduction approaches such as topological optimization provide a solution to determining the optimal shape of a structure without compromising the intended performance and required functionality. For most of the structures, topology optimization problems have been formulated to either minimize the compliance or weight of a structure under volume or stress constraints, respectively subjected to the certain boundary and loading conditions. In the present work, the optimization is performed by minimizing the compliance, subjected to a volume constraint, results in weight-saving of the railway freight bogie bolster without affecting the strength. A penalization scheme, SIMP (Solid Isotropic Material with Penalization) method is used to determine the optimum distribution of material, and void has been incorporated in the initial design. Topology optimization run performed on the initial design and results suggest mass reduction from chosen design space. The initial freight bogie bolster design is modified resulting in an approximate weight reduction of about 6.23%. Stress and deformation study performed by the FEA tool on the modified design of bolster justifies the optimization work. The modal analysis confirms the interchangeability of the designs.

A Topology Optimization Method Based on Non-Uniform Rational Basis Spline Hyper-Surfaces for Heat Conduction Problems

This work deals with heat conduction problems formulation in the framework of a CAD-compatible topology optimization method based on a pseudo-density field as a topology descriptor. In particular, the proposed strategy relies, on the one hand, on the use of CAD-compatible Non-Uniform Rational Basis Spline (NURBS) hyper-surfaces to represent the pseudo-density field and, on the other hand, on the well-known Solid Isotropic Material with Penalization (SIMP) approach. The resulting method is then referred to as NURBS-based SIMP method. In this background, heat conduction problems have been reformulated by taking advantage of the properties of the NURBS entities. The influence of the integer parameters, involved in the definition of the NURBS hyper-surface, on the optimized topology is investigated. Furthermore, symmetry constraints, as well as a manufacturing requirement related to the minimum allowable size, are also integrated into the problem formulation without introducing explicit constraint functions, thanks to the NURBS blending functions properties. Finally, since the topological variable is represented by means of a NURBS entity, the geometrical representation of the boundary of the topology is available at each iteration of the optimization process and its reconstruction becomes a straightforward task. The effectiveness of the NURBS-based SIMP method is shown on 2D and 3D benchmark problems taken from the literature.

A pseudo density topology optimization approach in nonlinear electromagnetism applied to a 3D actuator

A pseudo density topology optimization approach is introduced in nonlinear electromagnetism, which follows the SIMP (Solid Isotropic Material with Penalization) method. The adjoint approach is used to arrive at the sensitivity of the objective function. The elaborated optimization procedure includes nonlinearity of the ferromagnetic material. On the other hand, an applied linear characteristic of the ferromagnetic material results in a too small or wrong electromagnetic actuator dimension. Non-conforming grids are used to achieve a high accuracy by computing the discrete form. At the same time the number of finite elements decreases strongly. The implementation is applied to an electromagnetic actuator in 2D and 3D.

Proposition and analysis of strut and tie models for short corbels from techniques of topology optimization

Abstract Reinforced concrete short corbels are components characterized to represent typical conditions of geometrical and static discontinuity. In general, the classical bending theory is not valid for their design. With the strut and tie method, a model of a self-balanced truss, a strategy of representation of the principal stress flow appears as a representation of the trajectories of the main stresses in these components. Within the context of obtaining the strut and tie models, topology optimization is an indicated technique for an automated process. Combined with a numerical analysis based on finite elements, the SIMP (Solid Isotropic Material with Penalization) method formulation, which is defined with the criterion of minimum strain energy restricted by the volumetric fraction, is used for the development of the models with the ABAQUS® v. 6.14.1 software. Therefore, with the material distribution posterior to the optimization and the validation based on normative codes, it is demonstrated that the tool is effective in the development of strut and tie models.

Investigation on part consolidation for additive manufacturing with SIMP method

Additive manufacturing (AM) process comprises of different technologies which fabricate physical components from 3D CAD models by adding material layer-upon-layer. The unique capabilities of AM technologies are shape complexity, material complexity and hierarchical complexity. These enable new opportunities for customization of the product; fabricate intricate and complex shapes easier than the traditional method. Therefore, AM allows to consolidate the number of parts of an assembly to a single component and fabricate it. Though reducing part count in the assembly can avoid some assembly operations, the fullest potential of AM could not be achieved without material reduction. So the optimization step employed to lightweight the design. This paper demonstrates the above with an assembly of bactibiogramscope gantry head. Solid Isotropic Material with Penalization (SIMP) used to minimise material consumption. The optimised assembly was fabricated by the Fused Deposition Modelling (FDM) process. The final design was evaluated by comparing FEA result with experimental results. It also shows that topology optimized consolidate assembly take less time and material fabricate. This paper highlights the integration of part consolidation and topology optimizations to capitalize on the AM technique for fabricating better products.

Accelerating gradient-based topology optimization design with dual-model artificial neural networks

Topology optimization (TO) is a common technique used in free-form designs. However, conventional TO-based design approaches suffer from high computational cost due to the need for repetitive forward calculations and/or sensitivity analysis, which are typically done using high-dimensional simulations such as finite element analysis (FEA). In this work, artificial neural networks are used as efficient surrogate models for forward and sensitivity calculations in order to greatly accelerate the design process of topology optimization. To improve the accuracy of sensitivity analyses, dual-model artificial neural networks that are trained with both forward and sensitivity data are constructed and are integrated into the Solid Isotropic Material with Penalization (SIMP) method to replace the FEA. The performance of the accelerated SIMP method is demonstrated on two benchmark design problems namely minimum compliance design and metamaterial design. The efficiency gained in the problem with size of 64 × 64 is 137 times in forward calculation and 74 times in sensitivity analysis. In addition, effective data generation methods suitable for TO designs are investigated and developed, which lead to a great saving in training time. In both benchmark design problems, a design accuracy of 95% can be achieved with only around 2000 training data.

An 89-line code for geometrically nonlinear topology optimization written in FreeFEM

Topology optimization has emerged as a powerful tool for structural configuration design. To further promote the development of topology optimization, many computer programs have been published for educational purposes over the past decades. However, most of the computer programs are constructed based on a linear assumption. This paper presents an 89-line code for nonlinear topology optimization written in FreeFEM based on the popular SIMP (solid isotropic material with penalization) method. Excluding thirteen lines which are used for explanation, only 76 lines are needed for the initialization of the design parameters, nonlinear finite element analysis, sensitivity calculation, and updated design variables. Different design problems can be solved by modifying several lines in the proposed program. The complete program is given in the Appendix and is intended for educational purposes only.