Solid Isotropic Material Penalization Research Articles

Multiscale design based on non-penalization smooth-edged material distribution for optimizing topology (SEMDOT)

This paper presents a concurrent topology optimization method for macro and micro phases based on non-penalization smooth-edged material distribution for optimization topology (SEMDOT) method. Although there is existing research on the multiscale design method, grayscale elements are always emerged especially for penalization method for example the solid isotropic material penalization (SIMP) method, also high computational cost are required when large scale of elements are utilized for obtaining high resolution structures. The methodology proposed here aims to apply a new tech called non-penalization SEMDOT method to find the optimum layout on both scales of elements, it is assumed that the macro structure is composed of periodic materials and both element scales are optimized through their linearly interpolated grid points. The effective macroscopic properties are evaluated by the homogenization method. The approach could provide smooth and clear boundaries for multiscale system without grayscale elements or high computational cost. A series of numerical examples are presented to demonstrate the effectiveness and the efficiency of the proposed method.

Design of 3D-printed prostheses for reconstruction of periacetabular bone tumors using topology optimization.

Background: Prostheses for the reconstruction of periacetabular bone tumors are prone to instigate stress shielding. The purpose of this study is to design 3D-printed prostheses with topology optimization (TO) for the reconstruction of periacetabular bone tumors and to add porous structures to reduce stress shielding and facilitate integration between prostheses and host bone. Methods: Utilizing patient CT data, we constructed a finite element analysis (FEA) model. Subsequent phases encompassed carrying out TO on the designated area, utilizing the solid isotropic material penalization model (SIMP), and this optimized removal area was replaced with a porous structure. Further analyses included preoperative FEA simulations to comparatively evaluate parameters, including maximum stress, stress distribution, strain energy density (SED), and the relative micromotion of prostheses before and after TO. Furthermore, FEA based on patients' postoperative CT data was conducted again to assess the potential risk of stress shielding subsequent to implantation. Ultimately, preliminary follow-up findings from two patients were documented. Results: In both prostheses, the SED before and after TO increased by 143.61% (from 0.10322 to 0.25145mJ/mm3) and 35.050% (from 0.30964 to 0.41817mJ/mm3) respectively, showing significant differences (p < 0.001). The peak stress in the Type II prosthesis decreased by 10.494% (from 77.227 to 69.123MPa), while there was no significant change in peak stress for the Type I prosthesis. There were no significant changes in stress distribution or the proportion of regions with micromotion less than 28μm before and after TO for either prosthesis. Postoperative FEA verified results showed that the stress in the pelvis and prostheses remained at relatively low levels. The results of follow-up showed that the patients had successful osseointegration and their MSTS scores at the 12thmonth after surgery were both 100%. Conclusion: These two types of 3D-printed porous prostheses using TO for periacetabular bone tumor reconstruction offer advantages over traditional prostheses by reducing stress shielding and promoting osseointegration, while maintaining the original stiffness of the prosthesis. Furthermore, in vivo experiments show that these prostheses meet the requirements for daily activities of patients. This study provides a valuable reference for the design of future periacetabular bone tumor reconstruction prostheses.

Structural and thermal generative design using reinforcement learning-based search strategy for additive manufacturing

This paper proposes a reinforcement learning-based topology optimization (TO)/generative design approach using the Upper Confidence Bound (UCB) method to achieve a diverse set of optimal part designs for additive manufacturing. The UCB method has been integrated into two density-based TO problems – a compliance minimization problem and a thermal conduction problem along with Design for Additive Manufacturing (DfAM) filters to ensure improved additive manufacturability of the resultant topologies using different AM processes. The DfAM constraints are enforced by applying the support minimization filter and the thin feature minimization constraint in the TO model. The Solid Isotropic Material Penalization based TO model is perturbed to various levels of exploration and exploitation using the UCB exploration parameter to present different optimal designs. Unlike other data-reliant deep learning methods used in TO, the non-data-driven learning method proposed in this research is based on historical TO iterations integrated into a DfAM-constrained TO model, and it improves the method’s scalability to real-world and often computationally expensive part design applications. This paper fulfills a gap in computationally efficient methods for exploring generative designs of structural and thermal loaded parts with improved functional performance and additive manufacturability.

Experimental investigations of optimized 3D Printing Planar X-joints manufactured by stainless steel and high-strength steel

Welded joints are often employed in high-rise buildings and grid-shell structural systems but on-site welding of cut steel tubes for spatially complex frame systems is problematic, time consuming and difficult to ensure quality. Innovative prefabrication combining topology optimization (TO) using the Solid Isotropic Material Penalization Method (SIMP) and additive manufacturing (AM) provides an innovative solution for materially efficient mass production of repetitive joints. Optimized AM joints can be bolted to other members on-site, which promotes construction efficiency, and overcomes the barriers of on-site welding. In this paper, tensile tests are conducted to assess the material properties of mild and stainless steel materials used in different manufacturing processes. Welded and 3D printed joints using different materials are then investigated to maximize the mechanical properties by taking full advantage of AM and TO. A customized non-contact 3D-DIC (Digital Image Correlation) technique based on an open source tool (MultiDic) is applied to analyze and visualize the strain distribution in planar tubular and topologically optimized joints. AM joint exhibits a uniform stress distribution which avoids stress concentrations and the innovative configuration of the optimized joint has good energy absorption resulting in the protection of the central core region of the joint.

Fused Deposition Modeling and Characterization of Heat Shape Memory Poly(lactic) Acid-Based Porous Vascular Scaffold.

Shape memory polymers have received widespread attention from researchers because of their low density, shape variety, responsiveness to the environment, and transparency. This study deals with heat-shape memory polymers (SMPs) based on polylactic acid (PLA) for designing and fabricating a novel porous vascular scaffold to treat vascular restenosis. The solid isotropic material penalization method (SIMP) was applied to optimize the vascular scaffolds. Based on the torsional torque loading of Hyperworks Optistruct and the boundary conditions, the topological optimization model of a vascular scaffold unit was established. Forward and reverse hybrid modeling technology was applied to complete the final stent structure's assembly. The glass transition temperature for the present SMPs is 42.15 °C. With the increase in temperature, the ultimate tensile strength of the SMPs is reduced from 29.5 MPa to 11.6 MPa. The maximum modulus at room temperature was around 34 MPa. Stress relaxation curves show that the material classification is a "thermoset" polymer. The superb mechanical properties, the transition temperature of the SMPs, and the recovery ratio made it a feasible candidate for a vascular scaffold. A circular tube based on the shape memory polymers was presented as an example for analyzing the recovery ratio in an unfolding state. A higher recovery ratio was obtained at a temperature of 65 °C with a tube thickness of 2 mm. Finally, the proposed porous vascular scaffold was successfully fabricated, assessed, and compared with the original and previously developed vascular scaffolds. The proposed scaffold structure regains its initial shape with a recovery ratio of 98% (recovery temperature of 47 °C) in 16 s. The tensile strength, Young's modulus, and bending strength of the proposed scaffold were 29.5 MPa, 695.4 MPa, and 6.02 MPa, respectively. The results showed that the proposed scaffold could be regarded as a potential candidate for a vascular implantation.

Topology optimization design for buckling analysis related to the size effect

Owing to the excellent performance of microstructures or nanomaterials with well-designed topological configuration, the characteristic scale of structural design is gradually shifting from macroscopic to nanoscale or microscale structural design. However, the size effect that emerges from the small-scale structures may not be explained effectively with the hypothesis of classical mechanics owing to the lack of microscopic parameters in the classical constitutive model. In addition, slender beams within such small-scale structures are prone to buckling failure, which puts forward additional requirements for the stability design of the structure except for the overall compliance of the structure. Therefore, a topology optimization framework combining the modified couple stress theory with the solid isotropic material penalization (SIMP) model is constructed to illustrate the size effect on topology optimization. Numerical results show that the size effect affects the compliance, buckling performance, and topological configurations of the evolutionary structures.

Modification of the Stamp Topological Optimization Taking into Account Cyclic Fatigue based on the Finite Element Approach

The study is devoted to optimizing the volume of stamping tools used in pressure processing processes. The relevance of the research is due to the active development of additive technologies and the possibility of producing stamping tools from plastic of optimal shape, which has an important practical significance in the manufacture of thin-walled products in the aviation and automotive industries. The purpose of the study was to carry out a mathematical formulation of the problem of topological optimization of a forming die made of a polymer material with restrictions on fatigue durability and minimum volume. The task of topological optimization was to maximize the stiffness of the die under multicyclic loading. The vector description of topological optimization was based on the finite element approach. The optimization model was built on the basis of the solid isotropic material penalization method with the introduction of additional restrictions in the model of searching for pseudo-densities of the material, taking into account the duration of the force action on the stamp under multicyclic loading. In view of the nonlinearity of the resulting system of equations, the solution of the conditional optimization problem is proposed to be carried out by constructing the Lagrange objective function and using the Lagrange multiplier method. The result of the study is the proposed approach to the topological optimization of the stamp, taking into account the multicyclic loading and restrictions on the desired volume.

Improvement in 3D topology optimization with h‐adaptive refinement using the Cartesian grid Finite Element Method

AbstractThe growing number of scientific publications on topology optimization (TO) shows the great interest that this technique has generated in recent years. Among the different methodologies for TO, this article focuses on the well‐known solid isotropic material penalization (SIMP) method, broadly used because of its simple formulation and efficiency. Even so, the SIMP method has certain drawbacks, namely: lack of precision in definition of the edges of the optimized geometry and final results strongly influenced by the discretization used for the finite element (FE) analyses. In this article, we propose a combination of techniques to limit the effect of these drawbacks and, thus, to improve the behavior of TO. All these techniques are based on the use of the Cartesian grid finite element method (cgFEM), an immersed boundary method whose Cartesian grid structure and hierarchical data structure makes it specially appropriate for TO. All the proposed techniques are framed under the concept of mesh refinement. First, we propose the use of two meshes, the FE analysis mesh, and a finer mesh for integration and evaluation of sensitivities, to improve the resolution of the final solution at a marginal computational cost. Then we propose two h‐adaptive mesh refinement strategies. The first one will tend to refine the elements having intermediate density values and will have the effect of sharpening the definition of the edges of the optimized geometry. We will clearly show that if the accuracy of the FE analyses is not taken into account, stress constrained TO will generate solutions that, once manufactured, will not satisfy the constraints. Hence, we also propose an h‐refinement strategy based on the estimation of the discretization error in energy norm.

Topology optimization for three‐dimensional elastoplastic architected materials using a path‐dependent adjoint method

AbstractThis article introduces a computational design framework for obtaining three‐dimensional (3D) periodic elastoplastic architected materials with enhanced performance, subject to uniaxial or shear strain. A nonlinear finite element model accounting for plastic deformation is developed, where a Lagrange multiplier approach is utilized to impose periodicity constraints. The analysis assumes that the material obeys a von Mises plasticity model with linear isotropic hardening. The finite element model is combined with a corresponding path‐dependent adjoint sensitivity formulation, which is derived analytically. The optimization problem is parametrized using the solid isotropic material penalization method. Designs are optimized for either end compliance or toughness for a given prescribed displacement. Such a framework results in producing materials with enhanced performance through much better utilization of an elastoplastic material. Several 3D examples are used to demonstrate the effectiveness of the mathematical framework.

Multi-Material Topology Optimization Using Variable Density Lattice Structures for Additive Manufacturing

Multi-material lattice structures are used in a range of load-bearing applications for multiple conditions including mechanical and thermal loads. Additive manufacturing processes with multi-material capabilities are well suited to manufacture multi-material structures. In this paper, a multi-material topology optimization approach has been presented using variable-density lattice structures where the geometry of the lattice structure is pre-defined. The objective of the proposed topology optimization method is to design lightweight parts with minimized compliance and thermal energy or improve the heat transfer capability. To facilitate that, a novel interpolation scheme based on the stiffness matrices of the lattice structures has been proposed. This interpolation scheme, unlike the traditional Solid Isotropic Material Penalization (SIMP) interpolation, is observed to perform better in terms of approximating the structure’s load-bearing capacity, primarily due to its formulation on the lattice’s stiffness matrices. This cubic Hermite spline-based interpolation scheme makes it amenable for gradient-based optimization methods. A sequential linear programming method has been used to solve the weighted multi-objective optimization model. A Pareto-frontier study has also been carried out to fully characterize the trade-offs between the two objectives – compliance minimization and thermal energy minimization.

Extruded-geometric-component-based 3D topology optimization

A three-dimensional (3D) topology optimization approach based on extruded geometric components (EGCs) is proposed. Each EGC is constructed by extruding a convex/non-convex polygon along the axis of the EGC and rounding the ends of the EGC. Using an adaptive mapping technique which allows mapping each ECG onto a support domain, the EGCs are mapped onto an analytical grid to obtain an effective density field for material interpolation. Moreover, 2D-plane calculations can be utilized to replace 3D-space calculations to enhance computing efficiency. The positions and the cross-sectional areas of the ECGs are simultaneously optimized through the determination of an optimum set of geometry parameters. Some structural benchmark problems were investigated to verify the applicability of the proposed approach. Compared with the solid isotropic material penalization (SIMP) approach, the underlying approach does not require any filtering or projection techniques. Hence, it can produce a stiffer optimum design with an explicit boundary description whilst the number of design variables dramatically reduces.

Inverse homogenization using isogeometric shape optimization

Finding periodic microstructures with optimal elastic properties is usually tackled by a highly resolved, regular finite element model and solid isotropic material penalization. This procedure not only has many advantages, but also requires a comparably high computational effort and challenges in representing stresses accurately. Therefore, an isogeometric shape optimization approach is applied to the inverse homogenization problem and combined with a reconstruction procedure for nearly optimal rank-3 laminates, which provides an efficient solution strategy with more accurate stress modelling. This allows to investigate the sensitivity of optimized microstructures in terms of stress concentrations.

Time-dependent topology optimization of bone plates considering bone remodeling

Bone plates have been widely used for the treatment of bone defects and trauma. These fixation plates can stabilize or replace bone tissue to restore appropriate load-bearing functionality. Nevertheless, the use of bone plates may lead to the stress shielding, thereby weakening prosthetic bone substitutes (e.g. bone graft or scaffolds) due to significant change in the biomechanical environment after implantation. To address this issue, we propose a time-dependent topology optimization procedure for the design of bone plates by taking into account bone remodeling. A solid isotropic material penalization (SIMP) model is used to interpolate design variables. The objective is to maximize total bone density within a reconstruction area at the final stage of bone remodeling, subject to a volume constraint of the bone plate and maximum allowable compliance of the prosthetic system. The sensitivity of bone density at the final stage is derived with respect to the topological variables of the plate in a step-wise manner. To facilitate sensitivity analysis, a bone remodeling rule is formulated in two different ways to accommodate a C1continuity. A jaw reconstruction problem is exemplified in this study to demonstrate the effectiveness of the proposed approach. Through this specific case, the non-differentiability issue due to the lazy zone of a remodeling rule is smoothed; and the proposed approach is also compared with that of a time-independent design. The effects of volume fraction and compliance constraints are also investigated to gain further insights into the design of prosthetic substitutes. Together with additive manufacturing technology, the proposed time-dependent topology optimization procedure is expected to form a useful tool for the design of implantable devices ensuring favorable long-term treatment outcomes.

Design of a Bi-stable Airfoil with Tailored Snap-through Response Using Topology Optimization

This study aims to harness the geometric non-linearity of structures to design a novel camber morphing mechanism for a bi-stable airfoil using topology optimization. The goal is to use snap-through instabilities to actuate and maintain the shape of the morphing airfoil. Topology optimization has been used to distribute material over the design domain and to tailor the nonlinear response of the baseline structure to achieve the desired bi-stable behavior. The large scale deformation undergone by the structure is modeled using a hyperelastic material model. The non-linear structural equilibrium equations are solved using arc-length and displacement-controlled Newton–Raphson (NR) analysis. Isoparamteric finite element evaluation is used for analyzing kinematic and deformation characteristics of the structure. The optimization problem is solved using a computationally efficient nonlinear optimization algorithm, the Method of Moving Asymptotes (MMA), with a Solid Isotropic Material Penalization (SIMP) scheme. The gradient information required for the optimization has been evaluated using an adjoint sensitivity formulation. Two different design domains, one with a structured quadrilateral mesh and other with an unstructured triangular mesh, are investigated and compared. The effect of different optimization parameters on the final optimized structure and its behavior has also been analyzed. The final result is a novel camber morphing mechanism without the disadvantages of increased weight and higher maintenance costs associated with conventional actuation mechanisms.

Generating support structures for additive manufacturing with continuum topology optimization methods

PurposeThe purpose of this paper is to explore the possibility of combining additive manufacturing (AM) with topology optimization to generate support structures for addressing the challenging overhang problem. The overhang problem is considered as a constraint, and a novel algorithm based on continuum topology optimization is proposed.Design/methodology/approachA mathematical model is formulated, and the overhang constraint is embedded implicitly through a Heaviside function projection. The algorithm is based on the Solid Isotropic Material Penalization (SIMP) method, and the optimization problem is solved through sensitivity analysis.FindingsThe overhang problem of the support structures is fixed. The optimal topology of the support structures is developed from a mechanical perspective and remains stable as the material volume of support structures changes, which allows engineers to adjust the material volume to save cost and printing time and meanwhile ensure sufficient stiffness of the support structures. Three types of load conditions for practical application are considered. By discussing the uniform distributive load condition, a compromise result is achieved. By discussing the point load condition, the removal work of support structures after printing is alleviated. By discussing the most unfavorable load condition, the worst collapse situation of the printing model during printing process is sufficiently considered. Numerical examples show feasibility and effectiveness of the algorithm.Research limitations/implicationsThe proposed algorithm involves time-consuming finite element analysis and iterative solution, which increase the computation burden. Only the overhang constraint and the minimum compliance problem are discussed, while other constraints and objective functions may be of interest.Practical implicationsCompared with most of the existing heuristic or geometry-based support-generating algorithms, the proposed algorithm develops support structures for AM from a mechanical perspective, which is necessary for support structures particularly used in AM for mega-scale construction such as architectures and sculptures to ensure printing success and accuracy of the printed model.Social implicationsWith the rapid development of AM, complicated structures result from topology optimization are available for fabrication. The present paper demonstrates a combination of AM and topology optimization, which is the trend of fabricating manner in the future.Originality/valueThis paper remarks the first of attempts to use continuum topology optimization method to generate support structures for AM. The methodology used in this work is theoretically meaningful and conclusions drawn in this paper can be of important instruction value and practical significance.

A Novel Design Framework for Structures/Materials with Enhanced Mechanical Performance.

Structure/material requires simultaneous consideration of both its design and manufacturing processes to dramatically enhance its manufacturability, assembly and maintainability. In this work, a novel design framework for structural/material with a desired mechanical performance and compelling topological design properties achieved using origami techniques is presented. The framework comprises four procedures, including topological design, unfold, reduction manufacturing, and fold. The topological design method, i.e., the solid isotropic material penalization (SIMP) method, serves to optimize the structure in order to achieve the preferred mechanical characteristics, and the origami technique is exploited to allow the structure to be rapidly and easily fabricated. Topological design and unfold procedures can be conveniently completed in a computer; then, reduction manufacturing, i.e., cutting, is performed to remove materials from the unfolded flat plate; the final structure is obtained by folding out the plate from the previous procedure. A series of cantilevers, consisting of origami parallel creases and Miura-ori (usually regarded as a metamaterial) and made of paperboard, are designed with the least weight and the required stiffness by using the proposed framework. The findings here furnish an alternative design framework for engineering structures that could be better than the 3D-printing technique, especially for large structures made of thin metal materials.

Multi-Objective Structure Optimization Design on the Upper Carriage of a Naval Gun

A multi-objective structure optimization design onthe upper carriage of a naval gun was operated in order to improve both compliance and eigenfrequencies effectively. Based on SIMP (solid isotropic material penalization) method, compromise programming method was adopted to define multi-objective topology optimization. The topological structure of upper carriage was obtained through the optimization, and further, the new structure design. Results verified by FEA show that the new design can simultaneously satisfy the compliance and eigenfrequencies objective, and meanwhile candecrease mass and stress.

Multi-Objective Structure Optimization Design of a Car Lower Control Arm

A multi-objective structure optimization design of a car lower control arm was operated in order to improve both compliance and eigenfrequencies effectively. Based on SIMP (solid isotropic material penalization) method, compromise programming method was adopted to define multi-objective topology optimization. The topological structure of lower control arm was obtained through the optimization, and further, the new structure design. Results verified by FEA show that the new design can simultaneously satisfy the compliance and eigenfrequencies objective, and can meet yield stress requirements.

Study on Topology Optimum Design of Middle Pan

At present the study on the application of the topology optimum design in scraper conveyor is still under development. In this paper the topology optimization based on the Solid Isotropic Material Penalization (SIMP) is applied to the design of the middle pan, which can provide a valuable conceptual design scheme for the selection of the middle pan pattern. In conjunction with the homogenization theory and Finite Element Method (FEA), the mathematical models for designing the middle pan by maximizing the rigidity are developed in this paper. By applying the topology optimum design to the middle pan, the topology optimization based on the SIMP method is proved to be reliable and practical at the conceptual design stage of middle pan.

A Uniform Optimum Material Based Model for Concurrent Optimization of Thermoelastic Structures and Materials

This paper presents an optimization technique for structures composed of uniform cellular materials in macro scale. The optimization aims at to obtain optimal configurations of macro scale structures and microstructures of material under certain mechanical and thermal loads with specific base material volume. A concurrent topology optimization me- thod is proposed for structures and materials to minimize compliance of thermoelastic structures. In this method macro and micro densities are introduced as the design variables for structure and material microstructure independently. Penalization approaches are adopted at both scales to ensure clear topologies, i.e. SIMP (Solid Isotropic Material Penalization) in micro- scale and PAMP (Porous Anisotropic Material Penalization) in macro-scale. Optimizations in two scales are integrated into one system with homogenization theory and the distribution of base material between two scales can be decided automati- cally by the optimization model. Microstructure of materials is assumed to be uniform at macro scale to reduce manufactur- ing cost. The proposed method and computational model are validated by the numerical experiments. The effects of tem- perature differential, volume of base material, numerical parameters on the optimum results are also discussed. At last, for cases in which both mechanical and thermal loads apply, the configuration of porous material can help to reduce the system compliance.