Topological Shape Optimization Research Articles

Review on structural optimization techniques for additively manufactured implantable medical devices

Recent developments in additive manufacturing (AM) have led to significant opportunities in the design and fabrication of implantable medical devices due to the advantages that AM offers compared to conventional manufacturing, such as high customizability, the ability to fabricate highly complex shapes, good dimensional accuracy, a clean build environment, and reduced material usage. The study of structural design optimization (SDO) involves techniques such as Topology Optimization (TO), Shape Optimization (SHO), and Size Optimization (SO) that determine specific parameters to achieve the best measurable performance in a defined design space under a given set of loads and constraints. Integration of SDO techniques with AM leads to utmost benefits in designing and fabricating optimized implantable medical devices with enhanced functional performance. Research and development of various lattice structures represents a powerful method for unleashing the full potential of additive manufacturing (AM) technologies in creating medical implants with improved surface roughness, biocompatibility, and mechanical properties. Furthermore, the integration of artificial intelligence (AI) and machine learning (ML) in structural optimization has expanded opportunities to improve device performance, adaptability, and durability. The review is meticulously divided into two main sections, reflecting the predictability of the implant’s internal structure: (a) unpredictable interior topology, which explores topology-based optimization techniques, and (b) predictable inner topology, concentrating on lattice structures. The analysis of the reviewed literature highlights a common focus on addressing issues such as stress shielding, osseointegration enhancement, customization to individual needs, programmable functionalities, and weight reduction in implant designs. It emphasizes significant advances in reducing stress shielding effects, promoting osseointegration, and facilitating personalized implant creation. The review provides a detailed classification of optimization methods, with each approach scrutinized for its unique contribution to overcoming specific challenges in medical implant design, thus leading to more advanced, effective, and patient-oriented implantable devices.

Topological Shape Optimization Design of the Whole Bead of 265/35R18 Steel-Belted Radial Tire

The tire bead, as the most important load-bearing component at the bead area, is closely related to the durability of the tire, but its structure is developing slowly. For this reason, the topological whole bead design was proposed, although it performs well, many defects existed due to the design based on traditional experience. Therefore, this paper studies the topology shape optimization algorithm, delves into the main criterion based on von Mises and the interlaminar shear stress, and provides guidance for the structurally optimal design of the 265/35R18 radial tire whole bead. The finite element simulation results show that the von Mises of the inner end of the chafer and the end of the carcass cord are reduced by 14.48% and 24.12%, respectively. The interlaminar shear stress decreased by 28.96% and 49.51%, respectively. The von Mises of chafer and carcass cord decreased by 13.17% to 40.36% and 7.71% to 20.51%, respectively. The optimization design is of great significance to further improve the safety performance of tires.

Co-design Optimization of Moving Parts for Compliance and Collision Avoidance

Design requirements for moving parts in mechanical assemblies are typically specified in terms of interactions with other parts. Some are purely kinematic (e.g., pairwise collision avoidance) while others depend on physics and material properties (e.g., deformation under loads). Kinematic design methods and physics-based shape/topology optimization (SO/TO) deal separately with these requirements. They rarely talk to each other as the former uses set algebra and group theory while the latter requires discretizing and solving differential equations. Hence, optimizing a moving part based on physics typically relies on either neglecting or pruning kinematic constraints in advance, e.g., by restricting the design domain to a collision-free space using an unsweep operation. In this paper, we show that TO can be used to co-design two or more parts in relative motion to simultaneously satisfy physics-based criteria and collision avoidance. We restrict our attention to maximizing linear-elastic stiffness while penalizing collision measures aggregated in time. We couple the TO loops for two parts in relative motion so that the evolution of each part’s shape is accounted for when penalizing collision for the other part. The collision measures are computed by a correlation functional that can be discretized by left- and right-multiplying the shape design variables by a pre-computed matrix that depends solely on the motion. This decoupling is key to making the computations scalable for TO iterations. We demonstrate the effectiveness of the approach with 2D and 3D examples.

Data-driven non-intrusive shape-topology optimization framework for curved shells

Aiming at obtaining an optimal configuration of curved shells in the compact design space, it is an effective way to perform the shape and topology optimization simultaneously. Considering the complexity in solving the coupled shape and topology optimization problem, this paper presents an easy-to-implement data-driven non-intrusive shape-topology optimization framework for curved shells, including two steps. In the off-line step, the shape equation-driven mesh deformation method is firstly proposed, contributing to using a small number of design variables for fast shape change and optimization of the skin of curved shell without re-modeling and re-meshing. Latin hypercube sampling is used to generate sample points as the input data in the design space, and the corresponding skin shape is obtained. In each design domain determined by different skin shapes, the topology optimizations are performed in parallel or by multiple optimization solvers, after that above multi-source optimization results are collected into the output dataset. The mapping relationship is trained between the input data and output data via surrogate modeling technique, and this non-intrusive concept contributes to integrating the independent shape and topology optimization into a single-loop data-driven optimization. The cross validation-Voronoi adaptive model updating method is used to improve the prediction accuracy. In the on-line step, the single-loop data-driven optimization is carried out using covariance matrix adaptive evolution strategy and adaptive optimization updating method, which can obtain optimized results more efficiently than direct topology optimization or shape optimization method. In order to verify the effectiveness of the proposed data-driven non-intrusive shape-topology optimization framework, three optimization examples are carried out. Example results show that compared with the results by the single topology optimization, the strain energy results by the proposed framework can be reduced by 20.55%, 15% and 51.6%, respectively, highlighting the outstanding optimization ability of the proposed framework.

Shape and topology optimization method for fiber placement design of CFRP plate and shell structures

In this study, we present a novel non-parametric shape-topology optimization method for fiber placement and orientation design, aiming at controlling the stiffness of CFRP (Carbon Fiber Reinforced Plastics) shell structures. The sum of squared error norm for achieving the target displacements on the arbitrarily specified position is minimized under the length constraints as a design problem. A simultaneous shape and topology optimization problem is formulated as a distributed-parameter optimization problem based on the variational method. The shape gradient function and the density gradient for this design optimization problem are theoretically derived with the Lagrange multiplier method, the material derivative method and the adjoint variable method. The solid isotropic material with penalization (SIMP) method is employed for topology optimization. The gradient functions derived are applied to the H1 gradient method for fibers to determine the optimal shape and topology. With the present method, the optimal fiber placement and orientation can be obtained without any parameterization. Numerical examples are solved to show the validity of the proposed method and the results are discussed.

Broadband acoustic lens design using gradient-based optimization and adjusting radii and positions of scatterers

In this talk, we present the semi-analytical gradient-based optimization (GBO) technique for an efficient design of the broadband acoustic lens. This idea differs from earlier inverse designs that use topology optimization, shape optimization, and generic algorithms. We derived a formula for the gradients of the absolute pressure at the focal point with respect to positions and radii of a set of cylindrical scatterers. The derived analytic form of gradients of absolute pressure complements the modeling when combined with optimization and parallel computing. The GBO algorithm maximizes the sound amplification at focal point by evaluating pressure derivative with respect to the cylinder positions and radii, and then perturbatively optimizing the position and radius of each cylinder in the lens while considering acoustic multiple scattering between the cylinders. Computations are performed using the MultiStart Global optimization solver with fmincon while supplying the gradient of pressure at the focus and the gradient of nonlinear geometrical constraints. The GBO of the broadband acoustic lens is illustrated performing several performance measures for the dependency on the wavenumber and the incidence angle. The method is presented giving numerical examples for non-uniform configurations of the cylindrical structures of various radii and material submerged in water.

Equivalent Modeling and Verification of a High-Steepness and Lightweight Elliptical Aluminum Mirror

In order to realize the compact layout of aerospace payloads, the design and manufacture of high-steepness lightweight aluminum alloy mirrors is a key technology to be explored. For high-steepness mirrors, the traditional method is to establish the initial thickness that satisfies the bending stiffness through finite element optimization iteration, which cannot achieve fast modeling and performance estimation. In this paper, firstly, the equivalent modeling method of the mirror with high steepness is proposed to achieve the equivalent of the elliptic mirror with a diameter of 410 × 310 mm and F# less than 0.7. Based on the mathematical model, topology shape optimization was used to build a highly lightweight mirror structure that could be quickly assembled, and the equivalent area–mass density of the mirror is less than 34 kg/mm2. Next, the rationality of design feasibility was verified by simulation analysis. Finally, by using single point diamond turning combined with post polishing process, the high-precision manufacturing of conventional aluminum alloy mirror was realized. The results show that the mirror shape accuracy is 1/10 λ (λ = 632.8 nm), and the surface roughness Ra is 3.342 nm. This research provides strong theoretical support and application prospects for the low-cost and rapid manufacturing of high-steepness lightweight aluminum alloy mirrors.

Singular geometry perturbation based method for shape-topology optimization in unsteady Stokes flow

This paper concerns the topological derivatives and its applications for solving shape and topology optimization problems in fluid mechanics. The fluid flow is governed by the unsteady incompressible Stokes equations in the two dimensional case. We derive a topological sensitivity analysis for this parabolic-type operator. The proposed approach is based on a preliminary estimate describing the variation of the velocity field caused by the presence of a small obstacle inside the fluid flow. We obtain a topological asymptotic expansion for the unsteady Stokes operator valid for a large class of shape functions and an arbitrarily shaped geometric perturbation. Then, the topological gradient is exploited for building an efficient and accurate topology optimization algorithm. Finally, we present some numerical investigations showing the efficiency of the proposed approach.

Allying topology and shape optimization through machine learning algorithms

Structural optimization is part of the mechanical engineering field and, in most cases, tries to minimize the overall weight of a given design domain, subjected to functionality constraints given in terms of stresses or displacements. The most relevant techniques are topology and shape optimization. Topology optimization provides the optimal material distribution layout into a given, static, design domain. On the other hand, shape optimization provides the optimal combination of the parameters that define the required parametrization of the domain's boundary. Both techniques have strengths and weaknesses, thus a hybrid optimization approach that combines the former techniques will define a more general structural optimization framework that will take advantage of their synergistic combination. The difficulty arises when communicating both techniques for which, in this paper, we propose a machine learning-based methodology. • We proposed a methodology that allies topology and parametrized shape optimization algorithms, obtaining an hybrid optimization algorithm. • We studied the strengths and weaknesses of topology and shape optimization algorithms, which justified its combination to obtain a synergetic effect. • The use of manifold learning techniques, such as the Locally Linear Embedding (LLE), to extract the principal geometrical modes from a set of material distributions that topology optimization algorithm provides. • In order to properly use the LLE technique, the quasi-boolean material distribution provided by the topology optimization algorithm must be transformed into a richer level-set with the distance information. • The geometrical modes, extracted with the LLE algorithm, may be used to generate brand new geometries, as they define a parametric geometrical model. • The parametric space of geometrical modes may be considered by a shape optimization algorithm as its design space.

Simultaneous shape and topology optimization method for frame structures with multi-materials

In this study, we present a novel non-parametric shape-topology optimization method for frame structures with multi-materials, aiming at designing more light and stiff frame structures. The sum of squared error norm for achieving the target displacements on the specified members is minimized under the volume constraints of multi-materials as a design problem. A simultaneous shape and topology optimization problem is formulated as a distributed-parameter optimization problem based on the variational method. The shape gradient function and the material gradient functions for this design optimization problem are theoretically derived with the Lagrange multiplier method, the material derivative method and the adjoint variable method. The generalized solid isotropic material with penalization (GSIMP) method is employed to classify into the multi-materials in topology optimization. The gradient functions derived are applied to the unified H1 gradient method for frame structures to determine the optimal shape and material variations. With this method, the optimal free-form and topology for arbitrary large-degree of design freedom frame structures with multi-materials can be obtained without any shape and topology parameterization. Numerical examples with various materials and different boundary conditions are demonstrated and the results are discussed.

Topology optimization of transient nonlinear heat conduction using an adaptive parameterized level-set method

ABSTRACT In this article, topology optimization of transient nonlinear heat conduction problems is solved by a parameterized level-set method. Nonlinear density-based and continuum shape sensitivity analyses are conducted for topology generation and shape optimization. Meanwhile, the backward time transient adjoint structures are derived. To suppress dependence of the initial guess for the parameterized level-set-based topology optimization, the solid isotropic material with penalization method is adopted to generate the initial topology design. Then, the parameterized level-set method is further performed to obtain the optimal shape. The material volume is strictly controlled by restricting the coefficients of radial basis functions within a dynamic bound, which can avoid incorrect topological results due to topological changes. Finally, three numerical examples are solved with the parameterized level-set method, which demonstrates that the proposed approach can obtain complex topological details with smooth boundaries for transient nonlinear heat conduction problems.

Sequentially coupled gradient-based topology and domain shape optimization

A coupled topology and domain shape optimization framework is presented that is based on incorporating the shape design variables of the design domain in the Solid Isotropic Material with Penalization topology optimization method. The shape and topology design variables are incrementally updated in a sequential fashion, using a staggered numerical update scheme. Non-Uniform Rational B-Splines are employed to parameterize the shape of the design domain. This not only guarantees a highly accurate description of the shape boundaries by means of smooth basis functions with compact support, but also enables an efficient control of the design domain with only a few control points. Furthermore, the optimization process is performed in a computationally efficient way by applying a gradient-based optimization algorithm, for which the sensitivities can be computed in closed form. The usefulness of the coupled optimization approach is demonstrated by analyzing several benchmark problems that are subjected to different types of initial conditions and domain bounds. The variation in simulation results denotes that a careful construction of the initial design domain is necessary and meaningful.

Blade Structure Design Based on Topology Optimization and Shape Optimization

Blade Structure Design Based on Topology Optimization and Shape Optimization

Design of the solar panel frame of the spacecraft by 3D printing with composite materials

The optimal design of the skeleton panel of the solar panels of the spacecraft was designed taking into account the increased physical and mechanical characteristics through the use of composite materials and the method of topological shape optimization.

Topology Optimization of Micro-Structured Materials Featured with the Specific Mechanical Properties

Micro-structured materials consisting of an array of microstructures are engineered to provide the specific material properties. This present work investigates the design of cellular materials under the framework of level set, so as to optimize the topologies and shapes of these porous material microstructures. Firstly, the energy-based homogenization method (EBHM) is applied to evaluate the material effective properties based on the topology of the material cell, where the effective elasticity property is evaluated by the average stress and strain theorems. Secondly, a parametric level set method (PLSM) is employed to optimize the microstructural topology until the specific mechanical properties can be achieved, including the maximum bulk modulus, the maximum shear modulus and their combinations, as well as the negative Poisson’s ratio (NPR). The complicated topological shape optimization of the material microstructure has been equivalent to evolve the sizes of the expansion coefficients in the interpolation of the level set function. Finally, several numerical examples are fully discussed to demonstrate the effectiveness of the developed method. A series of new and interesting material cells with the specific mechanical properties can be found.

Velocity field level‐set method for topological shape optimization using freely distributed design variables

SummaryThe velocity field level‐set topological shape optimization method combines the implicit representation in the standard level‐set method and the capabilities of general mathematical programming algorithms in handling multiple constraints and additional design variables. The key concept is to construct the normal velocity field using basis functions and the velocity design variables at specified points (referred to as velocity knots) in the entire design domain. In this study, the velocity design variables are decoupled from the level‐set grid points. Making use of this property, we can adaptively change the arrangement of the velocity knots as the structural boundary evolves. This provides more design freedom in the optimization and allows for a significant reduction in the number of design variables. Several numerical examples in two‐ and three‐dimensional design domains are presented to demonstrate the robustness and efficiency of the proposed method. We also show that changing the number of velocity knots may implicitly exert certain control on topological complexity and length scale.

Structural design optimization of knee replacement implants for Additive Manufacturing

Abstract Recent developments in additive manufacturing (AM) have led the way to extraordinary opportunities in the development and fabrication of surgical implants due to advantages that AM offers. The study of structural design optimization (SDO) involves strategies such as topology optimization (TO), shape optimization, and size optimization to achieve a desired functionality for a given set of loads and constraints while optimizing specific qualities such as the structure weight or uniform stress distribution. Thus, integration of structural design optimization (SDO) and additive manufacturing (AM) is a powerful way for designing and fabricating lightweight medical implants that replicate the biomechanical properties of the host bones, and minimize stress shielding related problems. This study is focused in proposing a setup of a proper methodology for the rapid development of optimized surgical implants. A tibia intramedullary implant for an 8-year old osteosarcoma patient is designed and optimized, through TO in Abaqus/Tosca, to reduce the weight of the implant and minimize stress shielding related problems. A weight reduction of about 30 % was achieved from structural design optimization. The overall viability of the proposed design concept was validated using finite element analysis (FEA), and a stainless steel 316 L prototype was fabricated via SLM. After analysing results, in order to address osseointegration it is proposed that lattice structures to be incorporated in future work. In addition to that, there will be structural modifications for the implant to be able to adjust as the patient grows.

Combination of topology optimization and Lie derivative-based shape optimization for electro-mechanical design

This paper presents a framework for the simultaneous application of shape and topology optimization in electro-mechanical design problems. Whereas the design variables of a shape optimization are the geometrical parameters of the CAD description, the design variables upon which density-based topology optimization acts represent the presence or absence of material at each point of the region where it is applied. These topology optimization design variables, which are called densities, are by essence substantial quantities. This means that they are attached to matter while, on the other hand, shape optimization implies ongoing changes of the model geometry. An appropriate combination of the two representations is therefore necessary to ensure a consistent design space as the joint shape-topology optimization process unfolds. The optimization problems dealt with in this paper are furthermore constrained to verify the governing partial differential equations (PDEs) of a physical model, possibly nonlinear, and discretized by means of, e.g., the finite element method (FEM). Theoretical formulas, based on the Lie derivative, to express the sensitivity of the performance functions of the optimization problem, are derived and validated to be used in gradient-based algorithms. The method is applied to the torque ripple minimization in an interior permanent magnet synchronous machine (PMSM).

Topological Derivatives of Shape Functionals. Part III: Second-Order Method and Applications

The framework of topological sensitivity analysis in singularly perturbed geometrical domains, presented in the first part of this series of review papers, allows the asymptotic expansion of a given shape functional with respect to a small parameter that measures the size of singular domain perturbations, such as holes, cavities, inclusions, source terms and cracks. This new concept in shape sensitivity analysis generalizes the shape derivatives from the domain boundary to its interior for admissible domains in two and three spatial dimensions. Therefore, the concept of topological derivative is a powerful tool for solving shape–topology optimization problems. There are now applications of topological derivative in many different fields of engineering and physics, such as shape and topology optimization in structural mechanics, inverse problems for partial differential equations, image processing, multiscale material design and mechanical modeling including damage and fracture evolution phenomena. In this second part of the review, a topology optimization algorithm based on first-order topological derivatives is presented. The appropriate level-set domain representation method is employed within the iterations in order to design an optimal shape–topology local solution. The algorithm is successfully used for numerical solution of a wide class of shape–topology optimization problems.

Topological Shape Optimization Based on Harmony Search Method

Topological Shape Optimization Based on Harmony Search Method