Homogenization-based Topology Optimization Research Articles

Evaluation of Neural Network-based Derivatives for Topology Optimization

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

Efficient inverse-designed structural infill for complex engineering structures

Inverse design of high-resolution and fine-detailed 3D lightweight mechanical structures is notoriously expensive due to the need for vast computational resources and the use of very fine-scaled complex meshes. Furthermore, in designing for additive manufacturing, infill is often neglected as a component of the optimized structure. In this paper, both concerns are addressed using a so-called de-homogenization topology optimization procedure on complex engineering structures discretized by 3D unstructured hexahedrals.Using a rectangular-hole microstructure (reminiscent to the stiffness optimal orthogonal rank-3 multi-scale) as a base material for the multi-scale optimization, a coarse-scale optimized geometry can be obtained using homogenization-based topology optimization. Due to the microstructure periodicity, this coarse-scale geometry can be up-sampled to a fine single-scale physical geometry with optimized infill, with only a minor loss in structural performance and at a fraction of the cost of a fine-scale solution. The upsampling on 3D unstructured grids is achieved through stream surface tracing, aligning with the optimized local orientations. The periodicity of the physical geometry can be tuned, such that the material serves as a structural component and also as an efficient infill for additive manufacturing designs.The method is demonstrated through three examples of varying geometrical complexity. It achieves comparable structural performance to “brute force” state-of-the-art methods but stands out for its significant computational time reduction. By allowing multiple active layers, the mapped solution becomes more mechanically stable, leading to an increased critical buckling load factor without additional computational expense. The control of active layers also provides direct control over the internal structure, i.e., infill, ensuring that the infill is incorporated as a structural component and enhancing the manufacturability of the de-homogenization procedure. Furthermore, the proposed approach exhibits promising results, achieving volume fractions and weighted compliance values within 5% of the large-scale SIMP model, while demonstrating a computational efficiency improvement ranging from 10 times to over 250 times.

Analysis of additively manufactured flexible wing model

PurposeThis paper aims to assess the feasibility of additively manufactured wind tunnel models. The additively manufactured model was used to validate a computational framework allowing the evaluation of the performance of five wing models.Design/methodology/approachAn optimized fighter wing was additively manufactured and tested in a low-speed wind tunnel to obtain the aerodynamic coefficients and deflections at different speeds and angles of attack. The flexible wing model with optimized curvilinear spars and ribs was used to validate a finite element framework that was used to study the aeroelastic performance of five wing models. As a computationally efficient optimization method, homogenization-based topology optimization was used to generate four different lattice internal structures for the wing in this study. The efficiency of the spline-based optimization used for the spar-rib model and the lattice-based optimization used for the other four wings were compared.FindingsThe aerodynamic loads and displacements obtained experimentally and computationally were in good agreement, proving that additive manufacture can be used to create complex accurate models. The study also shows the efficiency of the homogenization-based topology optimization framework in generating designs with superior stiffness.Originality/valueTo the best of the authors’ knowledge, this is the first time a wing model with curvilinear spars and ribs was additively manufactured as a single piece and tested in a wind tunnel. This research also demonstrates the efficiency of homogenization-based topology optimization in generating enhanced models of different complexity.

Topology optimization of fibre reinforced polymer lattice structures for additive manufacturing

The material anisotropy induced by the related additive manufacturing (AM) extrusion process and the geometric errors introduced into the unit lattices by the stair-stepping artifacts were coupled in the topology optimization model to ensure the accuracy and reliability of the optimal design for the fibre reinforced polymer lattice structures. The material properties of the 3D printed carbon fibre reinforced polymers (CFRP) were firstly systematically investigated by extensive tensile tests aided by the digital image correlation (DIC) system, through which the microscale material anisotropy was clarified. Moreover, the accurate orthotropic material models were introduced into the as-fabricated mesoscale unit lattices with geometric errors to obtain the effective mechanical properties. Finally, mechanical performance of the macroscale structures was optimized by the integration of the ideal and as-fabricated unit lattices with the homogenization-based topology optimization method. The AM process-induced material anisotropy and geometric errors result in significant deviations of the mechanical property of the topologically optimized macrostructures and more accurate topology optimization results can be achieved considering the microscale material anisotropy and the geometric error in the mesoscale unit lattice. The results obtained in this investigation indicate that it is essential to comprehensively consider the AM process-induced microscale material anisotropy and mesoscale geometric errors in the topology optimization of fibre reinforced polymer lattice structures to improve the accuracy and reliability of the optimal design.

Experimental and Numerical Modal Characterization for Additively Manufactured Triply Periodic Minimal Surface Lattice Structures: Comparison between Free‐Size and Homogenization‐Based Optimization Methods

Homogenization‐based topology optimization (HMTO) is one of the most extensively used grading methods to generate functionally graded lattice structures (FGLs). However, it requires a precharacterization of the lattices, which is time‐consuming. As a remedy, free‐size optimization‐based graded lattice generation (FOGLG) is explored as an alternative method to generate the FGLs. This article builds on the authors’ previous work in which the HMTO and FOGLG approaches are studied to improve the dynamic characteristic of a design by using a single lattice type, namely, double gyroid (DG) structure. To show applicability of the proposed methods, different lattice types including diamond (D), gyroid (G), and I‐WP are employed to create FGLs herein. The frequency response analysis is performed, and the results from HMTO and FOGLG are compared in terms of their accuracy and efficiency. The optimized designs are then reconstructed by relative density mapping (RDM) and enhanced relative density mapping (ERDM) methods. The fabricated test samples made of cobalt–chromium using the direct metal laser melting (DMLM) technique are then experimentally validated using a laser vibrometer. The results reveal that HMTO and FOGLG can be used on the lattice types with a variety of configurations and relative densities.

Topology optimization with graded infill accounting for loading uncertainty

The optimal design of composite structures made of a solid phase and a given fraction of graded infill is addressed, using homogenization-based topology optimization and accounting for uncertainty in loading amplitude. A two-phase material law with void is implemented to control the amount of graded infill to be distributed, along with its admissible density range. Numerical homogenization is used to derive the macroscopic elastic properties of an isotropic and two orthotropic infills that are commonly used in additive manufacturing. A minimum weight problem is endowed with a set of deterministic displacement constraints that are equivalent to stochastic displacement enforcements in case of normal distributions of the amplitude of the applied forces. Sequential convex programming is adopted to solve the arising multi-constrained problem. Numerical simulations are performed to assess the proposed algorithm and point out peculiar features of the achieved optimal solutions with respect to layouts found in case of deterministic loads. When a fraction of graded infill is prescribed, coated structures are retrieved, whose shape may be remarkably affected by the selected type of lattice.

A Streamline-Guided Dehomogenization Approach for Structural Design

Abstract We present a novel dehomogenization approach for the efficient design of high-resolution load-bearing structures. The proposed approach builds upon a streamline-based parametrization of the design domain, using a set of space-filling and evenly spaced streamlines in the two mutually orthogonal direction fields that are obtained from homogenization-based topology optimization. Streamlines in these fields are converted into a graph, which is then used to construct a quad-dominant mesh whose edges follow the direction fields. In addition, the edge width is adjusted according to the density and anisotropy of the optimized orthotropic cells. In a number of numerical examples, we demonstrate the mechanical performance and regular appearance of the resulting structural designs and compare them with those from classic and contemporary approaches.

De-homogenization of optimal 2D topologies for multiple loading cases

This work presents an extension of the highly efficient de-homogenization method for obtaining high-resolution, near-optimal 2D topologies optimized for minimum compliance subjected to multiple load cases. We perform a homogenization-based topology optimization based on stiffness optimal Rank-N microstructure parameterizations to obtain stiffness optimal multi-scale designs on relatively coarse grids. To avoid relatively thin microstructure features, we regularize the design by introducing a material indicator field which results in well-defined widths and structural boundaries. In order to avoid singularities from the multiple load case problem, the orientations of the microstructures are further regularized. Subsequently, we derive a single-scale interpretation of stiffness optimal multi-scale designs on a fine grid using de-homogenization. The single-scale interpretation can be derived without costly postprocessing analysis on the fine grid, as an implicit boundary formulation is used.The effect of starting guesses is discussed, as they are non-trivial for Rank-N microstructures. Different numerical examples verify the performance of the inexpensive high-resolution solutions, both in comparison to the Rank-N based homogenization solutions, to equivalent density-based large-scale solutions, as well as to strict isotropic microstructure solutions. Depending on starting guesses, the approach consistently delivers structural performance values within a few percent of density-based large-scale solutions with a CPU time reduction factor of more than 300. Finally, we confirm that isotropic as well as orthogonal Rank-2 microstructure models are inferior to stiffness optimal anisotropic microstructure models for minimum compliance problems subjected to multiple load cases.

An approach for the concurrent homogenization-based microstructure type and topology optimization problem

This work proposes a methodology for the concurrent homogenization-based optimization of the type and the configuration of the microstructure of the structural domain, based on a list of pre-defined composite microstructures. The candidate microstructures are represented on this list by their homogenized mechanical properties, as predicted by means of the 3D homogenization theory. As such, each candidate property is tied to a specific microstructure -the one it has been derived from- and is conditioned on its topology, i.e. the geometric configuration of its unit cell. Similar to the standard discrete multi-material optimization problem (DMOP), in order to identify per element/patch in the structural domain the optimal microstructure from that list, weights are assigned to the candidate homogenized properties, with respect to which the final discrete microstructure type optimization problem (DMTOP) is posed. The topology of the optimal microstructure is determined by the volume constraint(s) imposed on one or more of its material components in the homogenization-based topology optimization problem (HTOP). The aim of this work is to combine the DMTOP with the HTOP in a unique mathematical framework in order to determine a unique microstructure type per element/patch in the structural domain, concurrently optimized in its topology. The proposed methodology is built step-by-step through the introduction of four microstructure types of two distinct constituent materials. Upon these auxiliary microstructures, the generalized concurrent homogenization-based topology and discrete microstructure type optimization problem (HTDMOP) is formulated for compliance minimization of the structure. Further, it is illustrated how the DMOP can be perceived as a DMTOP, and hence be combined in a similar fashion with the HTOP for the concurrent homogenization-based topology and material optimization (HTMO) of the structural domain. The paper concludes with demonstrating the developed methodology on the benchmark academic case study of the 3D Messerchmitt-Bölkow-Blohm (MBB) beam for the case where the auxiliary microstructures are considered as candidates for the domain.

Topology optimization of CFRP hierarchical pyramidal structures fabricated by additive manufacturing

The optimal structural hierarchy was introduced into the conventional pyramidal structure by integrating the homogenization-based topology optimization (HMTO) with elastically isotropic elementary lattices for further improvement of structural efficiency. The mechanical properties and elastic isotropy of representative truss–lattices composed of struts and plate–lattices consisting of closed-cell sheets were firstly evaluated, based on which the equivalent out-of-plane compression modulus of the hierarchical pyramidal structure was theoretically analysed and then compared with experimental and numerical results to systematically clarify the effects of lattice topology on the structural performance. Furthermore, the elementary lattice exhibiting the optimal structural efficiency and elastic isotropy was integrated with HMTO to optimize the out-of-plane compression properties including stiffness and energy absorption capability. The hierarchical pyramidal structure with optimal structural hierarchy achieves 4.8% and 28.2% higher compressive modulus over the conventional pyramidal structure and unoptimized hierarchical pyramidal structure, respectively. The energy absorption capability of optimized and unoptimized hierarchical pyramidal structures is 63.4% and 27.2% higher than that of conventional pyramidal structure without structural hierarchy, respectively. The results obtained from this study indicate that the structural hierarchy introduced via integration of isotropic SC–FCC plate–lattice with HMTO can effectively improve the structural efficiency of hierarchical pyramidal structures.

Topology optimization and de-homogenization of graded lattice structures based on asymptotic homogenization

This paper presents a novel two-step homogenization-based topology optimization and de-homogenization method for the design of graded lattice structures. The lattice orientation and material layout are first optimized for square base cells in the macro scale. Then by introducing the lattice stretching design variables of micro base cells, which bridge the base cell distortion with lattice stretch, the error residual of mapping functions is integrated with compliance formulation to form a novel mixed optimization formulation, concurrently optimizing structural performance and mapping functions. The advantage of this formulation is two-fold. First, the micro design space is relaxed from square base cells to rectangular ones so that performance improvement is further expected. Second, an excellent agreement, in both shape and performance, between the projected single-scale lattice structures with the homogenization results is secured, as compared to the frequently adopted post-process procedure of constructing single-scale lattices, where performance deviation could arise for specific microstructural patterns. With the optimized mapping functions, de-homogenization procedure is carried out to construct single-scale spatially graded lattice structures, where a simple filter-projection operation is proposed to obtain fine-scale smoothed boundaries from coarse-scale homogenization results with zig-zag boundaries. Several numerical examples are presented and compared with conventional post-process treatment results to show the validity of the proposed method, and different kinds of lattice patterns adopted in this work show its versatility for a broad range of lattice patterns.

On approaches for avoiding low-stiffness regions in variable thickness sheet and homogenization-based topology optimization

On approaches for avoiding low-stiffness regions in variable thickness sheet and homogenization-based topology optimization

An efficient 3D homogenization-based topology optimization methodology

Homogenization theory forms the basis for solving the topology optimization problem (TOP) formulated for designing composite materials. Homogenization is proved to be an efficient approach to effectively determine the equivalent macroscopic properties of the composite material. It relies on the assumption that the composite material presents a periodic pattern on a microstructural level; the simplest repeating unit of the microstructure, that if isolated represents exactly the macroscopic behaviour of the material, is called the $$unit\ cell$$ . Scope of homogenization is to determine the macroscopic (or else effective) properties of the non-homogeneous unit cell, that is, determine the properties of the unit cell as if it was composed by homogeneous material. In this study, a simple methodology is proposed where homogenization is implemented on a 3D lattice unit cell, with its radius being considered as the alternating parameter for the homogenization procedure; different values of the radius result to different unit cell configurations and hence, to different equivalent properties. A fitting process takes place in order to appropriately model the variations of the obtained effective properties w.r.t the design parameter. The corresponding, homogenization-based TOP is formed and the accuracy of the proposed methodology is assessed on several case studies.

Homogenization-based topology optimization integrated with elastically isotropic lattices for additive manufacturing of ultralight and ultrastiff structures

The integration of topology optimization with lattice structures has shown great potential for the additive manufacturing (AM) of lightweight structures with superior mechanical properties and multifunctional characteristics. To further improve the design manufacturability, structural efficiency, structural isotropy and computational efficiency, the homogenization-based topology optimization (HMTO) method was proposed to integrate with plate–lattices exhibiting superior mechanical properties and excellent elastic isotropy. The validity of the proposed method was demonstrated by comparing the optimized models with conventional models composed of truss–lattices and solid materials. Results show that the proposed method highly improves stiffness and energy absorption capability.

Inverse homogenization design of lattice structures without scale separation

Scale separation is often assumed in most homogenization-based topology optimization (TO) frameworks for design of material microstructures. This work goes beyond the mainstream TO contributions by abandoning the scale separation hypothesis. First, it puts to evidence the limits of the homogenization-based approach when the size of the Representative Volume Element (RVE) is not negligible with respect to the structure. Then, a re-localized scheme bridging the RVE and the structure is proposed to reproduce the microscopic fields, while the structure problem at the macroscopic scale is solved only based on the coarse mesh. Finally, numerical experiments show interesting results on 2D lattice structures within the proposed framework giving a hint towards a feasible realization of the finite-scale lattice structures with current resolution of additive manufacturing technologies. Reported results evidence that the present method can lead to the same topology and stiffness of the optimized structures as the reference solution when the number of unit cell is relatively large, while reducing the computational costs significantly.

Topology optimization of microlattice dome with enhanced stiffness and energy absorption for additive manufacturing

The synergy of additive manufacturing (AM) with topology optimization has become a useful method for developing ultralight, ultrastiff structures with high energy absorption capability. To improve the weight-specific stiffness and energy absorption capability of the conventional dome commonly used as the core of sandwich sheets, a new concept of filling the solid part of the dome with stretch-dominated microlattices is proposed. The optimal density distribution of microlattices is obtained by integrating the homogenization-based topology optimization method with the lattice structure. The compressive and bending stiffnesses of the optimized variable-density microlattice domes are demonstrated to be 41.8% and 33.7% higher than those of the conventional solid domes, while the energy absorption of the microlattice dome during compression and three-point bending is increased by 297.5% and 85%, respectively. Investigation of the cell size effect on the mechanical properties of the microlattice dome reveals that a larger cell size contributes more to the weight-specific stiffness and energy absorption capability at a given overall volume fraction constraint. The topology optimization and construction methods described in this paper are universal and can be used for the further development of ultralight, ultrastiff structures with arbitrary macro shapes with microlattices as constituent units.

Singularity aware de-homogenization for high-resolution topology optimized structures

Homogenization-based topology optimization has been shown to be effective but does not directly create mechanical structures. Instead, the method gives a multi-scale description of the optimized design, e.g., lamination thicknesses and directions. To obtain a realizable single-scale design, one can perform a subsequent de-homogenization step. This is done by converting the lamination directions to integrable vector fields from which it is possible to compute a parameterization of the domain. Unfortunately, however, singularities often make it impossible to find integrable vector fields that align with lamination directions. We present a short introduction to homogenization-based topology optimization followed by an overview of different types of singularities and how they impinge on the problem. Based on this, we propose a singularity aware de-homogenization pipeline, where we use a method for vector field combing which produces consistent labeling of the lamination directions but also introduces necessary seams in the domain. We demonstrate how methods from computer graphics can subsequently be used to compute the final parameterization from which the mechanical structure can easily be extracted. We demonstrate the method on several test cases.

De-homogenization of optimal multi-scale 3D topologies

This paper presents a highly efficient method to obtain high-resolution, near-optimal 3D topologies optimized for minimum compliance on a standard PC. Using an implicit geometry description we derive a single-scale interpretation of optimal multi-scale designs on a very fine mesh (de-homogenization). By performing homogenization-based topology optimization, optimal multi-scale designs are obtained on a relatively coarse mesh resulting in a low computational cost. As microstructure parameterization we use orthogonal rank-3 microstructures, which are known to be optimal for a single loading case. Furthermore, a method to get explicit control of the minimum feature size and complexity of the final shapes will be discussed. Numerical examples show excellent performance of these fine-scale designs resulting in objective values similar to the homogenization-based designs. Comparisons with well-established density-based topology optimization methods show a reduction in computational cost of 3 orders of magnitude, paving the way for giga-scale designs on a standard PC.

Optimal truss and frame design from projected homogenization-based topology optimization

In this article, we propose a novel method to obtain a near-optimal frame structure, based on the solution of a homogenization-based topology optimization model. The presented approach exploits the equivalence between Michell’s problem of least-weight trusses and a compliance minimization problem using optimal rank-2 laminates in the low volume fraction limit. In a fully automated procedure, a discrete structure is extracted from the homogenization-based continuum model. This near-optimal structure is post-optimized as a frame, where the bending stiffness is continuously decreased, to allow for a final design that resembles a truss structure. Numerical experiments show excellent behavior of the method, where the final designs are close to analytical optima, and obtained in less than 10 minutes, for various levels of detail, on a standard PC.

Natural frequency optimization of 3D printed variable-density honeycomb structure via a homogenization-based approach

It is well-known that the effective mechanical properties of cellular structures can be tuned by varying its relative density. With the advancement of 3D printing, variable-density cellular structures can be fabricated with high precision using this emerging manufacturing technology. Taking advantage of this unique ability to fabricate variable-density cellular structure, an efficient homogenization-based topology optimization method for natural frequency optimization is presented in this work. The method is demonstrated using a cantilevered plate with a honeycomb structure and is validated by detailed finite element analysis and experiment. It is shown that the optimal design can be fabricated by 3D printing and shows significant enhancement in natural frequency and reduction in weight.