Standard Topology Optimization Research Articles

Robust topology optimization considering part distortion and process variability in additive manufacturing

This paper presents a method to consider uncertainties in the distortion prediction of additive manufacturing processes within robust topology optimization. The random variable of the stochastic additive manufacturing process is the inherent thermomechanical strain, typically determined by process characterization experiments. The value of the inherent strain per se encompasses uncertainty due to differences between characterization and production geometries, uncontrolled process variations, hatching pattern choices, and other effects not captured in layer-by-layer part-scale additive manufacturing simulation. These effects can be represented by different inherent strains for each realized layer, therefore, model the variability in part distortion. Instead of employing a more detailed simulation approach, requiring significantly more process data and computation time, our method aims at generating a robust design in this setting, to obtain parts exhibiting reduced distortion regardless of uncertainties in distortion prediction. The formulation benefits from the superposition potential within the employed process simulation. For robust optimization, the expected part distortion and its estimated variance are included in the standard density-based topology optimization algorithm’s objective function. The effectiveness of the approach is demonstrated by simultaneously optimizing structural performance combined with a minimized additive manufacturing part distortion under uncertain process conditions.

Finite Element Analysis of Upper Limb Splint Designs and Materials for 3D Printing.

Three-dimensional (3D) printed splints must be lightweight and adequately ventilated to maximize the patient's convenience while maintaining requisite strength. The ensuing loss of strength has a substantial impact on the transformation of a solid splint model into a perforated or porous model. Thus, two methods for making perforations-standard approach and topological optimization-are investigated in this study. The objective of this research is to ascertain the impact of different perforation shapes and their distribution as well as topology optimization on the customized splint model. The solid splint models made of various materials have been transformed into porous designs to evaluate their strength by utilizing Finite Element (FE) simulation. This study will have a substantial effect on the designing concept for medical devices as well as other industries such as automobiles and aerospace. The novelty of the research refers to creating the perforations as well as applying topology optimization and 3D printing in practice. According to the comparison of the various materials, PLA had the least amount of deformation and the highest safety factor for all loading directions. Additionally, it was shown that all perforation shapes behave similarly, implying that the perforation shape's effect is not notably pronounced. However, square perforations seemed to perform the best out of all the perforation shape types. It was also obvious that the topology-optimized hand splint outperformed that with square perforations. The topology-optimized hand splint weighs 26% less than the solid splint, whereas the square-perforated hand splint weighs roughly 12% less. Nevertheless, the user must choose which strategy (standard perforations or topology optimization) to employ based on the available tools and prerequisites.

Process-Specific Topology Optimization Method Based on Laser-Based Additive Manufacturing of AlSi10Mg Components: Material Characterization and Evaluation

In the laser powder bed fusion process (PBF-LB), components are built up incrementally by locally melting metal powder with a laser beam. This process leads to inhomogeneous material properties of the manufactured components. By integrating these specific material properties into a topology optimization algorithm, product developers can be supported in the early phases of the product development process, such as design finding. For this purpose, a topology optimization method was developed, which takes the inhomogeneous material properties of components fabricated in the PBF-LB process into account. The complex pore architecture in PBF-LB components was studied with micro-computed tomography (µCT). Thereby, three characteristic regions of different porosity were identified and analyzed. The effective stiffness in each of these regions was determined by means of resonant ultrasonic spectroscopy (RUS) as well as finite element analysis. Afterward, the effective stiffness is iteratively considered in the developed topology optimization method. The resulting design proposals of two optimization cases were analyzed and compared to design proposals derived from a standard topology optimization. To evaluate the developed topology optimization method, the derived design proposals were additionally manufactured in the PBF-LB process, and the characteristic pore architecture was analyzed by means of µCT.

A new multi-scale topology optimization framework for optimal combinations of macro-layouts and local gradings of TPMS-based lattice structures

In this work, optimal combinations of macro-layouts and local gradings of triply periodic minimal surface (TPMS)-based lattice structures are obtained by using multi-scale topology optimization. A new innovative framework is proposed by using two density variables in each finite element. The first variable is the local relative lattice density and it sets the effective orthotropic elastic properties of the element, which in turn are obtained by using numerical homogenization of representative volume elements of the particular TPMS-based lattice structure of interest. The second variable is a standard topology optimization macro density variable, which defines if the element should be treated as a void or contain the graded lattice structure by letting this variable be governed by the rational approximation of material properties (RAMP) model. By using such density variables for all elements, the compliance is minimized by separately constraining the volume of lattice structure and the volume of macro-layout by using two independent constraints. For benchmarks in 3 D, it is demonstrated that the stiffness is increased significantly by including local grading of the lattice structure compared to using a constant lattice density. It is also demonstrated how ultra-lightweight designs can be generated using the multi-scale formulation, and how the optimal multi-scale solutions easily can be realized to printable stl-files by using implicit based geometry modeling. Finally, the new multi-scale topology optimization framework is utilized to generate an optimal design combination of macro-layout and local grading of frame-based Gyroid structure for the established GE-bracket benchmark.

Design of Selective Laser Melting (SLM) Structures: Consideration of Different Material Properties in Multiple Surface Layers Resulting from the Manufacturing in a Topology Optimization

Topology optimization offers a possibility to derive load-compliant structures. These structures tend to be complex, and conventional manufacturing offers only limited possibilities for their production. Additive manufacturing provides a remedy due to its high design freedom. However, this type of manufacturing can cause areas of different material properties in the final part. For example, in selective laser melting, three areas of different porosity can occur depending on the process parameters, the geometry of the part and the print direction, resulting in a direct interrelation between manufacturing and design. In order to address this interrelation in design finding, this contribution presents an optimization method in which the three porous areas are identified and the associated material properties are considered iteratively in a topology optimization. For this purpose, the topology optimization is interrupted in each iteration. Afterwards, the three areas as well as the material properties are determined and transferred back to the topology optimization, whereby those properties are used for the calculation of the next iteration. By using the optimization method, a design with increased volume-specific stiffness compared to a design of a standard topology optimization can be created and will be used in the future as a basis for the extension by a global strength constraint to maintain the maximum permissible stress and the minimum wall thickness.

Shape preserving topology optimization for structural radar cross section control

The purpose of this paper is to present a shape preserving topology optimization method to prevent the adverse effects of the mechanical deformation on the Radar Cross Section (RCS). The optimization will suppress the variation of RCS on the perfect conductor surface by structural design. On the one hand, the physical optics method is utilized to calculate the structural RCS, which is based on the surface displacement field obtained from the finite element analysis of the structure. The corresponding design sensitivities of topology optimization are derived analytically and solved by the adjoint method. On the other hand, the RCS variation and mechanical performance are taken into account simultaneously by extending a standard compliance-based topology optimization model. Two optimization formulations are discussed in an illustrative example, where the influences of upper limits of the compliance and the RCS variation are considered. Two more examples are further tested to show the ability and validity of the proposed optimization method.

An Approach for Geometrically Nonlinear Topology Optimization Using Moving Wide-Bézier Components With Constrained Ends

Abstract Standard moving morphable component (MMC)-based topology optimization methods use free components with explicitly geometrical parameters as design units to obtain the optimal structural topology by moving, deforming, and covering such components. In this study, we intend to present a method for geometrically nonlinear explicit topology optimization using moving wide-Bézier components with constrained ends. Not only can the method efficiently avoid the convergence issues associated with nonlinear structural response analysis, but it can also alleviate the component disconnection issues associated with the standard MMC-based topology optimization methods. The numerical investigations proposed in this work indicate that the proposed method allows us to obtain results in accordance with the current literature with a more stable optimization process. In addition, the proposed method can easily achieve minimum length scale control without adding constraints.

Multi-scale topology optimization for stiffness and de-homogenization using implicit geometry modeling

In this article, we demonstrate the state-of-the-art of multi-scale topology optimization for 3D structural design. Many structures designed for additive manufacturing consist of a solid shell surrounding repeated microstructures, so-called infill material. We demonstrate the performance of different types of infill microstructures, such as isotropic truss or plate lattice structures and show that the best results can be obtained using spatially varying and oriented orthotropic microstructures. Furthermore, we demonstrate how implicit geometry modeling using nTop platform can help to interpret these multi-scale designs as single-scale manufacturable designs (de-homogenization). More importantly, we demonstrate the small difference in performance between these multi-scale and single-scale designs through extensive numerical testing. The presented method is at least 3 orders of magnitude more efficient compared to standard density-based topology optimization, allowing for high-resolution 3D structures to be obtained on a standard workstation PC.

Design of a patient-specific mandible reconstruction implant with dental prosthesis for metal 3D printing using integrated weighted topology optimization and finite element analysis

The aim of this study was used a weighted topology optimization method to design a patient-specific mandibular implant for reconstruction and restoration of appearance in patients with severe mandibular defects. A finite element (FE) model was constructed and the defect region was defined from the unilateral first premolar to the second molar. The reconstruction implant included main body, fixation wing and dental prosthesis. Standard topology optimization was performed using stress constraint to identify optimal fixation wing structure (denoted as WOS) with solid core main body. Two independent optimal main body with internal beam supporting structures defined as WOSA and WOSO optimized from the WOS model under axial and oblique conditions were then obtained, respectively. Final optimal model (WBOS) was generated using a weighted topology optimization that considered 60% and 40% contributions of WOSA and WOSO models, respectively. The WBOS model was fabricated using metal 3D printing and fixed on the resting acrylonitrile butadiene styrene (ABS) bone to perform fracture testing. Stress concentration were found in the upper area connected to the main body of the mesial wing and corresponding maximum values under axial/oblique loads were reduced from 778/925 MPa of the WOS model to 764/720 MPa of the WBOS model. The reduction in percentage variations of weight between original (91.1 g) and final optimal (24.5 g) models was 73.14% for fabricated 3D printing models. The WBOS model also exhibited a higher resistant force (2163 N) when compared with the original model (1678 N). This study developed a design strategy with weighted topology optimization and fabrication for producing patient-specific implants using metal 3D printing. The obtained reconstruction implant can provide good biomechanical performance and recovery of appearance for oral rehabilitation.

Simulator-based training of generative neural networks for the inverse design of metasurfaces

Abstract Metasurfaces are subwavelength-structured artificial media that can shape and localize electromagnetic waves in unique ways. The inverse design of these devices is a non-convex optimization problem in a high dimensional space, making global optimization a major challenge. We present a new type of population-based global optimization algorithm for metasurfaces that is enabled by the training of a generative neural network. The loss function used for backpropagation depends on the generated pattern layouts, their efficiencies, and efficiency gradients, which are calculated by the adjoint variables method using forward and adjoint electromagnetic simulations. We observe that the distribution of devices generated by the network continuously shifts towards high performance design space regions over the course of optimization. Upon training completion, the best generated devices have efficiencies comparable to or exceeding the best devices designed using standard topology optimization. Our proposed global optimization algorithm can generally apply to other gradient-based optimization problems in optics, mechanics, and electronics.

Modified element stacking method for multi-material topology optimization with anisotropic materials

This article presents a modified element stacking method for anisotropic multi-material topology optimization. This method can transform standard multi-material topology optimization formulations into a series of equivalent single-material topology optimization ones to overcome the various limitations inherent to conventional techniques. First, typical multi-material topology optimization methods utilize material interpolation schemes that restrict the study to the isotropic domain with a constant Poisson’s ratio, limiting practical applications and solution accuracy. Additionally, in attempts to further extend these classical multi-material topology optimization methods to anisotropic materials, preparing the necessary element information matrices becomes increasingly laborious or even impossible for complex models, preventing the application of sensitivity analysis for robust gradient-based optimization methods. To directly address these limitations, the element interpolation method replaces material interpolation schemes with an element interpolation framework, where each design cell property is determined using a weighted sum of various coincident single-material elements. This paper provides a thorough description of element interpolation and presents several numerical examples demonstrating successful implementation.

Topology optimization of the multi-fasteners jointed structure considering joint load constraint and fatigue constraints

The purpose of this paper is to present an important strength issue in the design of multi-fasteners jointed structures. Multi-fastener joints such as bolts and rivets are widely used in assembled structures and always act as critical elements in damage tolerance design. Therefore, fasteners static strength and fatigue performance are simultaneously considered in this paper. On one hand, constraints on joint loads are issued to avoid the fasteners failure. On the other hand, two multiaxial fatigue criterion, i.e., Sines criterion and Crossland criterion are introduced to avoid the failure in the connection areas around fasteners. In addition, the total material usage is limited for the light-weight design. The standard topology optimization is thus extended to minimize the structural compliance with joint load constraints and fatigue failure constraints. Beam elements are used to model the fasteners. The joint loads can therefore be derived easily based on the finite element formulation. In order to eliminate the high stresses caused by the nodal connections of the beam elements, compositional modeling method is used to evaluate the stress level around the fasteners, in which beam elements are used to simulate the stiffness of fasteners and spring/shell elements are used to simulate the interaction of multi-fastener joints. Fatigue constraints are handled in the context of stress-based topology optimization while the corresponding analysis is based on static finite element due to the decomposition of harmonic loads. To address the singularity problems related to stress constraints, q - p relaxation is used and p ≥ q ≥1. Fatigue constraints should be evaluated in every element which inevitably brings in a large number of constraints. P-norm is therefore used as the constraints aggregation scheme. The aggregation factor of P-norm, i.e., Pn is assigned as 6 in this paper. With respect to the constraints aggregation process, the design sensitivities of fatigue constraints and joint load constraints are derived and calculated. Two numerical examples, namely an assembled I-shape beam and a cabin of the high-speed flight vehicle are tested. Optimized results with the proposed method are compared with standard topology optimization design. All the constraints are completely controlled by the prescribed upper level and the loss of stiffness in the optimized configuration can be seen as a trade-off for the strength. It is shown that the fatigue performance is strongly affected by the structural layout and the load carrying path. The effects of different fatigue constraints on the optimized configuration are studied.

Topology optimization of the multi-fasteners jointed structure considering fatigue constraints

This paper is to present an important issue of fatigue failure in the design of multi-fasteners jointed structure. To avoid failure in the connection area, Sines criterion is utilized. Fatigue constraints are handled in the context of stress based topology optimization. To eliminate the high stresses caused by the finite element modeling, the control volume is defined to evaluate the stress states around the fasteners. The standard topology optimization is extended to minimize the structural compliance with fatigue failure constraints. To address singularity problems related to stress constraints, q-p relaxation is used. P-norm is used as the constraints aggregation scheme. Basing on the above, the design sensitivity of fatigue constraints is derived and calculated. The proposed method is verified by a numerical example of an assembled I-beam. The comparisons of the numerical results have shown the effect of the fatigue constraint.

Multi-material topology optimization for automotive design problems

The algorithms for multi-material topology optimization were developed to solve compliance-minimization problems and applied to engineering problems in automotive concepts and lightweight design. Two small-scale problems of a long cantilever and a control arm were studied initially to verify the effectiveness of the developed algorithms and in-house program. Optimal solutions achieved by the multi-material topology optimization method developed were compared to their counterparts obtained by standard single-material topology optimization. To efficiently solve real-world engineering problems, the algorithms were further advanced to incorporate extrusion constraints and to handle multiple load cases. The effectiveness and the efficiency of the proposed method were demonstrated by the study of two real-world engineering problems: (a) the conceptual design of a cross-member for a chassis frame; and (b) the conceptual design of an automotive engine cradle. The two optimization design problems both involved complex geometries, design and non-design domains, prescribed regions with specific material allocations, multiple load cases, and manufacturing extrusion constraints. It was explicitly demonstrated that, for the same weight, the optimum designs achieved by the multi-material topology optimization method were stiffer than those achieved by standard single-material topology optimization.

Structural topology optimization for directional deformation behavior design with the orthotropic artificial weak element method

In this paper, the topology optimization method is extended to arrive at the desired deformation behavior within local structural domains by distinguishing and suppressing specific deformation in a certain direction. Compared to the standard topology optimization and existing shape preserving design method, the contribution of this paper is twofold. First, Artificial Weak Elements (AWE) are defined and attached to the appropriate local domain. The strain energies of the AWE are calculated as a measurement to describe the deformation behavior quantitatively. Second, particular orthotropic material properties are defined for the AWE according to the given direction of the desired deformation, which formulate Orthotropic Artificial Weak Elements (AWE ort ). By introducing the constraint on the strain energy of the AWE ort into a standard compliance based topology optimization, the deformation in the corresponding direction is suppressed, resulting in the desired deformation behavior. Several numerical examples are tested and compared to the standard topological design to illustrate the effect of the directional deformation behavior design.

Topology optimization of industrial robots: Application to a five-bar mechanism

Recent works introduced topology optimization in the design of robots, but the proposed methodologies led to a local optimization of the performance. Moreover, most of performance indices used are not in strong relation with easy-to-understand technological requirements.We propose a methodology that is able to perform a topology optimization for robots, valid globally in the workspace or for a set of given trajectories, and which is based on the use of technology-oriented performance criteria. In order to enforce the chosen performance indices to be valid globally, optimal robot configurations or trajectories for which extreme performance will be attained are computed, and iteratively updated.In order to decrease the computational time associated with these performance indices, we exploit the structure of the elastic models in order to reduce their computational complexity.Finally, we use an optimization algorithm called the Linearization Method which gives results in a computational time equivalent to standard topology optimization algorithms, but its implementation is less complex and makes it quite easy to perform modification or improvement.The methodology is applied for the design of a five-bar mechanism. We show that our approach leaded to a robust optimization of the robot performance over the whole workspace.

Stiffeners layout design of thin-walled structures with constraints on multi-fastener joint loads

The purpose of this paper is to present an extended topology optimization method for the stiffeners layout design of aircraft assembled structures. Multi-fastener joint loads and manufacturing constraints are considered simultaneously. On one hand, the joint loads are calculated and constrained within a limited value to avoid the failure of fasteners. On the other hand, the manufacturing constraints of the material distribution in the machining directions of stiffeners are implemented by an improved piecewise interpolation based on a beveled cut-surface. It is proven that the objective function is strictly continuous and differentiable with respect to the piecewise interpolation. The effects of the extended method with two different constraints are highlighted by typical numerical examples. Compared with the standard topology optimization, the final designs have clearly shown the layout of stiffeners and the joint loads have been perfectly constrained to a satisfying level.

Design and Manufacturing Functionally Gradient Material Objects With an Off the Shelf Three-Dimensional Printer: Challenges and Solutions

Abstract This paper presents the challenges and solutions encountered while designing and then printing functionally gradient material (FGM) objects using an off the shelf fused deposition modeling (FDM) 3D printer. The printer, Big Builder Dual-Feed Extruder from 3dprinter4u, Noordwijkerhout, The Netherlands, has the unique design of extruding two different filaments out of one nozzle. By controlling the rate at which the two filaments are pulled into the melt chamber, FGM objects can be printed. Software challenges associated with process planning required to print an FGM object are solved by showing a method for printing a discretized gradient and by designing an open-loop control mechanism for the extruder motors. A design method is proposed that models an object using a level-set function (LSF) with a material gradient. Instead of merely identifying the boundaries of the object, the level set also models the material gradient within the object. This representation method along with a genetic algorithm finds an optimal design for an FGM cantilever beam that is then printed on the FDM printer. The model and genetic algorithm are also used to solve a standard topology optimization problem. The results are compared to a similar FGM topology optimization method in the literature. All the codes for this paper are made open source to facilitate future research.

3503 リアクトルコアの磁場と構造の連成解析による最適防振設計(OS3-8 設計と最適化VIII)

This paper proposes optimal vibration-proof design of reactor core based on coupled analysis of magnetics and harmonic response. A reactor is a part of the in-vehicle DC-DC converter to boost the electric voltage, and it is composed of cores, coils, and so on. Because of electromagnetic and harmonic response characteristics of the reactor, the reactor core vibrates when the reactor core is in use, and as a result, noises are generated. To reduce these noises, a topology optimization method for reactor core design is proposed in this paper. Here, the main difficulty is to compute the electromagnetic force appropriately. Roughly speaking, standard topology optimization methods, such as density based methods, are not suitable to compute the electromagnetic force because grayscale elements smear out the structural boundary in which the electromagnetic force is imposed. Therefore, in this paper, boundary tracking type level-set based topology optimization is used where grayscale elements can be suppressed completely. Thanks to this grayscale-free topology optimization, the coupled analysis of magnetics and harmonic response is successfully performed, and as a result, topology optimization of the reactor core for reducing the noises is realized. The usefulness of the proposed method is confirmed using some numerical examples.

Structural topology optimization: Extensibility and attainability

Extensibility and attainability of topology optimization are discussed by investigating a variety of simultaneous topology opti-mization methods extended from the standard formulation. First, the state of the art is highlighted through systematic classification of developed methods, such as simultaneous topology optimizations of microstructure and macrostructure, structure and supports, structure and design-dependent loads, structure and locations of involved components. Second, some recent results about simultaneous topology optimization of structure and applied loads are presented. It is shown that the simultaneous topology optimization is an integrated methodology that extends the concept of standard topology optimization in the sense of systematic design. The presence of more than one kind of design variable of different nature makes the optimization problem complex but enlarges the design space to attain the optimization.