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Confirmed Speakers & Abstracts

Optimization Developments for large scale and non-linear modeling including CAD-reconstruction

Claus B.W. Pedersen1, Peter M. Clausen1, Kingshuk Bose2, Rong Fan2, Jaewon Jang2, Vladimir Belsky2, Anton Jurinic3, Pascal Hebrard4
1Dassault Systemes Deutschland GmbH, Germany; 2Dassault Systemes Simulia Corp., United States; 3Dassault Systemes AB, Sweden; 4Dassault Systemes SE, France

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The current work presents three topics on latest industrial solutions and workflows.

Recently, several advances have been done for the Abaqus Multi-Grid (AMG)-based iterative solver addressing large scale simulations with high performance and high accuracy. The AMG solution is implemented for both the primal systems of structural equations as well as for the adjoint systems for the sensitivities. Thus, the present solution addresses industrial large scale designing challenges, e.g. if one can 3D print a minimum size of 0.5 mm and the design space is 20x20x20 cm then a minimum finite element resolution of 64 million elements is required. For such models designers expect a relatively fast turnaround time and a relatively small amount of memory consumption. The presented AMG solution allows for realistic simulation and thereby, realistic optimization as unstructured and arbitrary meshes can be simulated including modelling of contact (penalty & augmented Lagrange method), gasket elements, pre-tension sections, tie constraints, MPC, kinematic couplings and rigid bodies, symmetry and periodicity boundary conditions, as well as a large number of boundary conditions.

Secondly, we demonstrate sizing designing for non-linear industrial optimization including material non-linearities, as well as geometrical non-linearities for large deformations and contacts. To the best of our knowledge, this is the first work that shows results for non-linear sizing of shell thicknesses and lattice radii using adjoint sensitivities, including simultaneously the three types of non-linearities.

Thirdly, we address one of the main obstacles in embedding topology optimization and other non-parametric optimization methods in the product design process, namely the lack of efficient and robust CAD- reconstruction methods. Here the focus is on semi-automated CAD-reconstruction of topology optimization results. Various strategies for CAD-reconstructions and for new CAD-features of industry based topology optimizations solutions will be presented in detail. Some of the new CAD reconstruction tools are focusing on free-form optimized structures for additive manufacturing whereas others are focused on more classic and traditionally manufactured methods, but still with a high degree of geometrical complexity matching the topology optimization results. The present solution combines classical CAD-solids and surfacing features in a single application that to the author's knowledge is the first dedicated workbench to CAD-reconstructed non-parametric optimization results.

For all above three topics several industrial design applications will be shown.

Research Needs to Transform the Topology Optimization into a Generative Design Tool

Andreas Vlahinos
Advanced Engineering Solutions

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The additive manufacturing (AM) industry continues to grow with new machines, faster processes and a large selection of materials. As design practitioners, we are rambling to unleash the full potential of AM. Lattice structures are very effective for lightweight structural panels, energy absorption devices, thermal insulation, ballistic protection and porous implants. A Taxonomy of Lattice Structures and an overview of the currently available generation techniques for on surface (i.e. hexagon), 2.5 D (i.e. honeycomb), 3D beam (i.e. diamond, octet) and 3D shell (i.e. gyroid, Lidinoid) lattice structures will be presented. An overview of the state of the art in commercially available Sub-Divisional surface modeling, topology optimization and lattice structure generation tools will be presented.

Although all these tools have been advanced over the years, there are not integrated parts of the product development process. The research needs and required enhancements to transform the topology optimization tools into functional generative design tools will be outlined.

Handling Assemblies with Contact in Generative Design

David Weinberg
Autodesk, Inc.

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For over a decade Generative Design has been remarkably popular in the engineering design community due to its capability on designing lightweight structures by optimally distributing materials to carry loads. Autodesk Nastran has implemented topology optimization (TO) since 2014 and rapidly enhancing its capabilities, including multidisciplinary optimization, hierarchical distributed computing, and various manufacturing method-oriented design. This presentation covers how Nastran TO handles complex assemblies with multiple design regions and nonlinear contact as well as welded regions. Each region can have its own stress allowable and material type. It is even possible to combine shell and solids elements in the same model. Issues with loads on non-preserve boundaries is also addressed.

Scalable Design for Manufacturing, Modeling Optimization for Additive Manufacturing

Devlin Hayduke1, Albert To2
1Materials Sciences LLC; 2University of Pittsburg

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With recent advances in additive manufacturing (AM) technologies, components and structures that can benefit from cellular design and optimization are now being realized. Materials Sciences LLC (MSC) has developed and demonstrated the design and analysis tools needed to optimize cellular (lattice) structures that exploit state-of-the-art manufacturing processes and tailor the frequency response of components and associated structures while minimizing weight. Unlike commonly available commercial tools for optimization of lattice structures, MSC's multi-scale optimization tools do not require a predetermined unit cell architecture or lattice network. Additionally, since the approach is based on well-established homogenization methods, the topology optimization based algorithms are numerically efficient. Using this approach, complex, cellular geometries only need to be explicitly reconstructed in a geometry format suitable for AM fabrication after the optimization; enabling designers and analysts to use conventional finite element analysis tools. Despite these advances, considerable research is required to develop robust structural components for missile structures. In particular, optimization of the strength of cellular structures and verification of their performance via experiments is needed. In response to this need, MSC's current research is focused on development of an integrated framework of design, analysis, and test and evaluation methodologies to design structural components using a lattice (cellular) design using AM processed titanium alloys. Particular attention is being paid to optimization of unit cell architectures and lattice networks for strength. This will be achieved by developing a physics based understanding of the modes of failure that can occur in a lattice networks, e.g., local yielding, local buckling, global buckling, etc., such that the strength of the cellular component can be optimized to meet performance objectives. By using a continuum approach, critical features, e.g., connection of lattice ligaments, that are often overlooked by commonly used tools that represent lattice networks using beam elements can be optimized for strength. Salient aspects of this research will be presented.

Enabling industrial research and development in level-set-based topology optimization

Julien Cortial1, Felipe Bordeu1, Chiara Nardoni2, Florian Feppon1, Charles Dapogny3, Grégoire, Allaire4
1Safran Tech; 2IRT SystemX; 3Univ. Grenoble-Alpes; 4 Ecole Polytechnique

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Topology optimization is routinely used in industry for the design of structural parts and many commercial off-the-shelf software solutions are available for this purpose. However some challenges remain, such as accounting for the specificities of the various additive manufacturing processes, tackling multi-physics (e.g. structural, thermal and flow) problems, or designing multi-scale structures. Unfortunately industrial research and development in topology optimization is hampered by the fact that virtually all commercial grade software packages are tightly coupled with a specific solver used to compute the state of the system, the values and the sensitivities of the design criteria.

This presentation focuses on an strategy that enables the rapid prototyping of level-set-based topology optimization while maintaining acceptable performance levels for challenging, realistic problems. First, a modular software architecture that decouples as much as possible the topology optimization engine from the physical solver is adopted. Second, the level-set method is combined with a remeshing procedure to maintain a body-fitted discretization during the optimization process. In the case of weakly coupled co-simulation, the different physics can be solved by dedicated, highly performant solvers that are not required to be level-set-aware. Also, this approach allows to model the physical behavior on the interface (conjugate convective heat transfer, flow boundary layer, etc.) more accurately.

Strength Prediction for Topology-Optimized Structure

Mahesh Chengalva, Robert Burkett, Tonfiz Mahmood, Ebonni Adams
Boeing

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As topology-optimized structures become more commonplace, there is a need for generalized methods of strength predictions for such structures. Due to the complexity of geometry, the application of classical 'hand analysis' methods is often difficult or impossible. Although conventional FEM analysis is able to incorporate the complex geometry and is well suited for predictions of displacement, stresses and strains, failure prediction can be a challenge on account of complex failure criteria especially for additively manufactured structures, where material properties vary with orientation and processing parameters.

To overcome these limitations, a FEM-based approach has been developed where the failure criteria as well as the material allowables are embedded within the FEM code. Complex models can then be loaded in any arbitrary manner, with the FEM software evaluating the state variables at all locations in the model continuously during loading. These state variables are then automatically utilized by the failure criteria to determine the occurrence of failure. Should a failure occur, the corresponding element is deleted from any further structural interaction. The same criteria continues to be applied so that progression of failure can be tracked by a series of initiations. Thus a simple set of rules that define material failure can potentially predict ultimate load carrying capability and the failure mode for geometrically and/or materially complex structures of all types (composites, plastics and metals).

This approach involves Explicit Finite Element analysis and the creation of special 'material subroutines' that encapsulate the failure criteria as well as basic strength allowables. Internally referred to us 'UFEM', this method has had widespread application at Boeing for conventional metallic structures used in aircraft. Utilizing only basic tensile and compressive stress strain curves as input, strength values for a variety of structures such as fittings, lugs, angle clips, channels among others has been predicted within 10% of the test data in most cases.

The ability to predict the structural strength prior to manufacture provides a significant advantage enabling design optimizations and trade studies to be run virtually, considerably reducing the size of the test matrix needed to physically test the part. Three case studies are presented over a range of topology-optimized structure used in aircraft, from metallic fittings to pressurized ducts to composites subjected to impact, where virtual testing capabilities have been correlated with test data, opening the possibility for expanding these methods to all types of topology optimized structure.

Topology Optimization of Nonlinear Structures with Bifurcating Paths: Origami Fold Pattern Design

Andrew Gillman1,2, Kazuko Fuchi2,3, Alexander Pankonien2, Philip Buskohl2
1UES Inc., 2Air Force Research Laboratory, 3University of Dayton Research Institute

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Mechanical instabilities are increasingly being leveraged for the design of material architectures with unique and advantageous metamaterial properties, such as negative stiffness, auxetic response and vibration mitigation. Key mechanisms for mechanical instability include buckling and snap-through bifurcations, which harness spatial mismatches in mechanical properties and geometric nonlinearities to generate rapid transitions between stable equilibrium configurations. Origami, the art of paper folding, possesses these requisite properties for instability design, given the large localized rotations and stiffness mismatch between folding and facet stretching/bending inherent in the structure. While certain origami fold patterns, e.g. the frequently studied Miura-ori pattern (rigidly foldable), have been shown to exhibit these advantageous mechanical metamaterial properties, an automated design framework taking into account multiple modes of deformation (facet stretching/bending and folding) has proven challenging. Beyond developing a robust design framework, the mechanics or origami structures can be highly nonlinear, and challenges include navigating the origami design space while considering the branching of multiple fold paths off of the flat state, which are often difficult to discriminate energetically.

To investigate these challenge, we combine a nonlinear truss model and topology optimization techniques to identify methods for robust fold path selection and fold pattern discovery. A nonlinear mechanical analysis method that balances efficiency and accuracy has been developed including techniques for exploring critical points and bifurcations. The nonlinear truss modelcaptures the geometric nonlinearities associated with large rotations and enables systematic study of how the stretch, bend and fold stiffness ratios affect the design space. A genetic algorithm, given its natural parallelizability for supercomputer deployment, has been utilized to explore several non-convex objective functions including target actuation motion, tuning of bi-stable energy wells, and auxetic material behavior. Current efforts are focused on navigating the design space for origami structures with many bifurcating paths while exploring the interplay of material stiffness distribution and geometric imperfections, and exploiting these features to bias one bifurcated equilibrium path over another within the optimization process. This automated design framework enables discovery of origami-based systems for diverse applications including energy absorbing materials, programmable compliant robots, space deployable antenna and solar arrays, and morphing and adaptive structures.

Topology optimization for multi-physics applications

Tsz Ling Elaine Tang1, Lucia Mirabella1, Songtao Xia1, Suraj Musuvathy1, Wentao Fu2
1Siemens Corporation, Corporate Technology; 2Siemens Energy, Inc.

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Topology optimization is an evolving technique for generating freeform designs. One classical example is the generation of a design that minimizes minimize structural compliance subject to a volume/weight constraint under given loading conditions. Typical loading conditions involve forces of fixed magnitude at fixed locations in the domain. In the past decades, new methods and new features (e.g. manufacturing constraints, multi-physics, multi-scale and novel optimization algorithm) have been developed to expand the capabilities of topology optimization, in order to apply this technology into real-world applications. Traditionally, the algorithm only considers structural loads, and cannot be used to solve design problems involving thermal and flow considerations. Such designs are usually subjected to different types of loading conditions (e.g. pressure, convection boundary conditions). In this work, a topology optimization framework is proposed to handle the requirements and complexities of multi-physics applications. Algorithms and strategies to handle the associated complexity will be presented. The new approach has the potential to generate novel designs for hot gas path components of gas turbines and aircraft engines that are often subject to harsh thermal conditions.

Practical Considerations for Topology-Optimized Additively Manufacturing Parts

Matt Lynch
United Technologies Research Center

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Topology optimization is a powerful design methodology for parts with structural, thermo-fluid and other functionality. Recent advances in additive manufacturing complement this design capability with the ability to produce a variety of different shapes having extremely complex form factors including lattices. Unfortunately, the surfaces of the parts tend to be rough, particularly in metal additive manufacturing, and can lead to a debit in fatigue properties which may be enough to sacrifice gains achieved by topology optimization. This presentation will discuss this issue and its implications for use of topology-optimized parts in aerospace. A potential solution for surface finishing of lattices will also be presented along with preliminary experimental results.

A Multi-layer Support Based Filter to Constrain the Minimum Overhang Angle in Topology Optimization for Additive Manufacturing

Hongyu Zhu1, Grum Ngatu2, Matt Lynch1
1United Technologies Research Center; 2Collins Aerospace

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Additive manufacturing (AM) allows the creation of components in a layer-by-layer, additive fashion, which offers enormous geometrical freedom compared to conventional manufacturing technologies. It is widely recognized that topology optimization is essential to exploit the design space AM allows. However, overhang limitation in additive manufacturing prevents the direct production of topology optimized parts. Post-processing is generally need. Lately, a layerwise filter has been incorporated in density-based topology optimization on uniform structured meshes for print-ready designs. With this technique, the minimum allowable overhang angle (the angle a down-facing surface has with the base plate) is restricted to 45 degree. In practice, smaller overhang angles cause more roughness. At times, a greater overhang angle is desired due to the smoothness requirement. In this work, we present a multi-layer based overhang constraint that allows minimum overhang angle greater than 45 degree without changing the element aspect ratio of the mesh. The newly developed constraint is demonstrated on 2D examples while it can be extended to 3D.

Optimization-based Design for Manufacturing

Joshua Robbins, Brett Clark, Miguel Aguiló
Sandia National Laboratories

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Advances in manufacturing continue to increase the available design space. While this creates exciting possibilities in realizable designs, it also creates a trade space between functional performance (design freedom) and manufacturability (design certifiability) that is difficult to navigate with current design tools. This talk will describe a new effort to create design software that will allow engineers to compute optimized designs that respect the limitations of various manufacturing processes and enable informed decisions about the trade-offs between functional performance and ease/cost of manufacture.

In the current AM workflow, parts are realized in two distinct stages - design and manufacture. Producers rely on highly-skilled technicians to successfully manufacture parts and provide feedback regarding the "printability" of designs. Iterations are typically required between the designer and AM expert to arrive at a design and associated process parameters that result in an acceptable scrap rate. This manual cycle is slow and relies on engineering heuristics to guide design modifications. Simulation-based software tools are beginning to appear in industry that help with the selection of process parameters to mitigate process limitations, but these tools do not directly influence the incoming design. The current effort is to integrate fast, accurate AM process simulation into optimization-based design to provide predictions of the distortion and residual stress during optimization. This will result in designs that accommodate the limitations of AM and are therefore optimally printable. In this talk, the details of the approach and progress towards process-informed optimization will be presented.

Cellular Level Set in B-Splines (CLIBS): A Parametric Level Set Method for Topology Optimization of Solid/Cellular Structures

Michael Yu Wang, Hongming Zong, Qingping Ma, Ye Tian
Hong Kong University of Science & Technology

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In this work, we develop a level set modeling technique for designing and optimization of solid/cellular structures, called cellular level set in B-Splines (CLIBS). In this technique, the entire design domain for the solid/cellular structure in question is subdivided into a set of connected volumetric cells in three dimensions. In the level set scheme for geometric representation of a structure, an implicit trivariate B-spline function is defined on each subdomain cell. This parameterization scheme allows us to impose constraints on the relevant B-spline coefficients for naturally maintaining geometric continuities at the connection faces between neighboring cells. Benefitting from the intrinsic properties of the trivariate B-spline functions, the method offers several useful properties and powerful functionalities to build and modify a solid/cellular structure in the modeling process and to conduct topology optimization. The topology optimization of solid/cellular structures is directly obtained in terms of the B-spline coefficients, with the use of a fast B-spline interpolation implemented as a sequence of discrete convolution operations. The CLIBS method completely eliminates the numerical difficulties typically associated with the classical global level set method, in particular, the finite difference scheme for solving the Hamilton-Jacobi equation, re-initialization, and velocity extension. Several 3D modeling and optimization examples are presented, including single-scale solid structures, periodic cellular structures and layered cellular structures. The proposed method is highly scalable, potentially leading to high definition modeling and optimization applications on a large-scale computing platform.

PLATO: Multidisciplinary design optimization platform

Miguel Aguiló, Joshua Robbins, Brett Clark
Sandia National Laboratories

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This talk presents Sandia's multidisciplinary design optimization platform. The new PLATO ecosystem provides a suite of modeling, analysis and optimization tools for generating smooth, connected designs across multiple hardware platforms. The core technologies are integrated with a full functioning graphical user interface that enables users to specify the design envelope and performance requirements, launch multiple, simultaneous parallel simulations and monitor the optimization progress. In addition, the new application programming interfaces allow plug-n-play insertion of non-native simulation tools. This intrinsic flexibility enables academic, industry and government partners to augment, leverage and experiment with PLATO. Finally, this talk will present multiple examples highlighting the range of solutions (multi-physics, uncertainty-awareness, multi-material) that are possible with the new PLATO engine as well as ongoing research & development projects.

Scalable Algorithms for High-resolution Topology Optimization

Graeme Kennedy
Georgia Institute of Technology

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Advancements in additive manufacturing have enabled the creation structures with geometric resolutions that exceed those achievable using state-of-the-art topology optimization algorithms. To close this gap, new topology optimization algorithms and implementations are required to realize the potential of additively manufactured structures. As a step towards this goal, our research combines both novel optimization algorithm development and large-scale topology optimization implementations. To achieve efficient large-scale topology optimization implementations, we utilize octree-based methods are highly scalable and well-suited for local mesh refinement in parallel computations. Recent application of this approach include the solution of large-scale stress and frequency-constrained topology optimization, and multimaterial optimization of large-scale structures with up to 329 million elements. Multimaterial problems often exhibit special constraint structure which must be exploited to achieve a scalable implementation. High-resolution can also be achieved through the use of higher-order methods. Recent developments in our group have enabled thermal elastic optimization of single and multimaterial structures with up to 6th order analysis elements, and 5th order design parametrizations. These methods yield smooth structural surfaces with fewer degrees of freedom or design variables than low-order methods. Finally, structures optimized for stiffness and strength may suffer from structural stability issues. To address this problem, we have developed novel methods for eigenvalue-constrained topology optimization with specific application to buckling-constrained design. These methods leverage inexpensive quadratic approximations of the eigenvalue constraint that provides additional curvature information to inform the optimization algorithm, enabling more robust eigenvalue-constrained optimization.

Universal Machine Learning for Topology Optimization

Heng Chi1, Yuyu Zhang1, Tsz Ling Elaine Tang2, Lucia Mirabella2, Glaucio H. Paulino1
1Georgia Institute of Technology; 2Siemens Corporation, Corporate Technology

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Topology optimization is a powerful computational tool that has been extensively implemented in several commercial software and successfully used in many industrial applications (e.g. aerospace and automotive). Because topology optimization is an iterative procedure, a major limitation is its computational expense. A topology optimization problem typically involves hundreds of design iterations, and in order to update the current design, the structural response needs to be solved to compute the sensitivity information. To handle large-scale topology optimization problems (e.g., problems involves millions of design variables and beyond), the associated computational cost could be enormous and one typically has to resort to parallel computing, advanced iterative solvers, or multi-scale and multi-resolution approaches.

In this work, we put forward a general machine learning-based topology optimization framework which greatly accelerates the design process of large-scale problems in 3D. The proposed framework has several novel features. First, unlike other machine learning-based framework in the topology optimization literature, the machine learning module of the proposed framework is trained simultaneously during the topology optimization. Second, the framework adopts a tailored discretization setup which enables a training strategy that is based on local data. The localized training strategy can improve both the scalability and accuracy of the proposed framework. Third, the framework incorporates an online update scheme which continuously reinforces the machine learning module to improve its accuracy. Finally, the proposed machine learning-based topology optimization framework is universal in the sense that it can work with any suitable machine learning model and has the potential to be applied in a wide range of topology optimization problems (minimum compliance, compliant mechanism, maximum thermal conductivity, and so on). Through numerical investigations and several design examples, we demonstrate that the proposed framework is highly scalable and can efficiently handle design problems with a wide range of discretization levels, different load and boundary conditions, and various design considerations.

Phase Diagrams in Topology Optimization

Dan White
Lawrence Livermore National Laboratory

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We are concerned with the computational topological optimization of elastic structures, in particular minimization of compliance subject to a constraint on the mass. Through computational experiments it is discovered that even very simple optimization problems can exhibit complex behavior such as critical points and bifurcation. In the vicinity of critical points structural topology optimization problems are not well-posed since infinitesimally small perturbations lead to distinct topologies. This is reminiscent of phase diagrams in materials science. In this talk it is argued that topology design problems have distinct phases, and a goal of topology optimization should be to construct the phase diagram of the design problem. Preliminary results on this topic will be presented.

Support Structure Design Optimization for Metal Additive Manufacturing Enabled by Modified Inherent Strain Method

Albert To1, Lin Cheng1, Xuan Liang1, Qian Chen1, Devlin Hayduke2, Ryan Whitmore2
1University of Pittsburgh; 2Materials Sciences Corporation

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Metal additive manufacturing (AM) as an emerging manufacturing technique has been gradually accepted by engineers to manufacture end-use components. However, one of the most critical issues preventing its broad applications is build failure resulting from residual stress and deformation inherent in the melting and solidification process in AM. This issue causes delamination and cracking during the manufacturing process, which can stop the recoater blade from spreading the powder, as well as lead to part warpage after removing from the building tray. To address these issues, a new design optimization method based on fast process simulation is proposed for the design of support structure, in order to reduce residual stress and distortion and ensure manufacturability. First, a modified inherent strain method is proposed and employed for fast prediction of the stress and deformation. It is based on thermomechanical modeling at mesoscale and implemented as a few static mechanical analyses, rather than a time-consuming transient solve. Thus, process simulation can be significantly accelerated and compatible with structural design. Second, a projection scheme is proposed to map the domain of support structure for a given solid component, in which the minimum support area is found for the next step. Third, lattice structure topology optimization is applied to minimize the mass consumption of support structure subjected to yield stress constraints. This not only prevent failure of the AM build by limiting the residual stress below the yield strength, but also reduce material required for support structure. Moreover, the self-support nature of lattice structure, and its ability to provide accurately mechanical representation, makes it a natural design tool for support structure design. Several examples are performed and manufactured to demonstrate the efficiency of the proposed algorithm. Both numerical simulation and experiments prove that the proposed method can significantly reduce residual stress and ensure the success of metal AM builds.

Design of Support Structures via Penalization of Fiber Orientation

Aaditya Chandrasekhar, Krishnan Suresh
University of Wisconsin-Madison

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In this paper, support structures for additive manufacturing (AM) are generated by posing a tailored thermal topology optimization problem. Specifically, we introduce auxiliary fibers, and allow for the concurrent evolution of the topology and fiber orientation. By penalizing the fiber orientation, we show that one can implicitly control the surface normal to satisfy overhang constraints. The formulation inherently offers a mechanistic interpretation to support structure design. The generality of the algorithm is demonstrated for a variety of scenarios and conditions in 2D and 3D.

Topology Optimization under Uncertainty using Stochastic Gradients

Subhayan De, Alireza Doostan, Kurt Maute
University of Colorado Boulder

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In topology optimization, uncertainties can arise from manufacturing processes and operating conditions, as well as from inadequacies of the computational models used to describe the physical system. To achieve a robust design, the effect of uncertainties should be considered in the formulation of the design objective and constraints. For topology optimization under uncertainty, we need to evaluate the forward model multiple times to accurately estimate the statistical moments of the objective and constraints at every design iteration. Similarly, multiple design sensitivity analyses need to be performed, further increasing the computational burden. To overcome this computational bottleneck, in this study stochastic gradient descent (SGD) algorithms are investigated. Stochastic approximations of the objective and constraints, as well as their design sensitivities, are estimated using only a few random samples per iteration. The results of the SGD algorithms are compared with the ones of the Globally Convergent Method of Moving Asymptotes (GCMMA) using stochastic gradients. Topology optimization with the Solid Isotropic Material with Penalization (SIMP) method and an explicit level set method are studied. Structural design problems in both two and three dimensions are solved in presence of uncertainties. These examples show that the use of a small number of random samples per iteration to estimate the objective, constraints and their design sensitivities leads to increase in cost by only a small factor compared to a deterministic topology optimization.

Stochastic Methods for Topology Optimization with Many Load Cases

Xiaojia Shelly Zhang1, Eric de Sturler2, Alexander Shapiro3, Glaucio H. Paulino3
1University of Illinois at Urbana-Champaign; 2Virginia Tech; 3Georgia Institute of Technology

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Practical engineering designs typically incorporate a large number of load cases. For topology optimization considering many deterministic load cases with a weighted-average formulation, a large number of linear systems of equations must be solved in each optimization step, leading to an enormous computational cost. In this work, we present stochastic methods that drastically reduce the total computational time. We reformulate the deterministic objective function and gradient into stochastic ones, which can then be estimated at the cost of a few iterative linear system solves, possibly as few as one. Note that the solution of this stochastic optimization algorithm solves the original deterministic optimization problem. The stochastic optimization method is combined with a damping strategy related to simulated annealing. Through numerical examples, we demonstrate that the use of the stochastic techniques allows us to solve large topology optimization problems at a drastically reduced computational cost while obtaining similar design quality. For example, for a simple 2D design problem, the number of linear solves could be reduced by a factor of about 40. The results indicate that the damping scheme is effective and leads to the rapid convergence of the proposed methods.

Topology optimization with local stress constraints: An aggregation-free approach

Oliver Giraldo-Londoño1, Fernando V. da Senhora1, Ivan F. M. Menezes2, Glaucio H. Paulino1
1Georgia Institute of Technology; 2Pontifical Catholic University of Rio de Janeiro

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We present a methodology for solving topology optimization problems with local stress constraints, which is based on the Augmented Lagrangian (AL) method. To improve the robustness of the method, both the penalty and objective function terms of the AL function are modified. The modification of the penalty term aims to reduce the mesh-dependency and that of the objective function term helps to drive the solution towards black and white (0/1). In addition, a variation of the vanishing constraint is used because it leads to results that outperform those obtained using relaxed stress constraints. Although maintaining the local nature of stress requires a large number of stress constraints, the formulation presented here requires only one adjoint vector whose cost of evaluation is similar to that of finding the global displacement vector. This elegant result leads to an efficient (inexpensive) sensitivity evaluation. Several 2D and 3D topology optimization problems, each with a large number of stress constraints, are solved to demonstrate the robustness of the proposed approach.

Topology Optimization for Brittle Fracture Resistance

Jonathan Russ, Haim Waisman
Columbia University

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Structural topology optimization in the context of material degradation and fracture has been gaining considerable interest in recent years. In light of this, we will present some recent work in which the phase field method for fracture is employed in order to increase the fracture resistance of a structure comprised of a quasi-brittle material in a density-based topology optimization setting. The phase-field approximation of the fracture surface energy is utilized in the optimization problem definition and analytical sensitivities are computed for the path-dependent problem. Numerical examples highlighting the effectiveness of the proposed approach will be presented and some more recent results will also be discussed.

Parameterized mapped microstructure for multiscale topology optimization

Tej Kumar, Krishnan Suresh
University of Wisconsin-Madison

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There has been a recent interest in High-resolution Topology Optimization (HTO), enabling unprecedented details in the design. However, the computational cost in HTO can be massive. An alternate approach is to rely on Multiscale Topology Optimization (MTO) where microstructures are optimized at a smaller scale, while simultaneously optimizing the structure at the macroscale. The two scales are linked through homogenization theory.In MTO, instead of optimizing microstructures along with macroscale optimization, we use parameterized single-void Vigdergauz-like microstructure and homogenize it for various values of these parameters first. Later, these parameters and a rotation variable are used for optimization of the structure for stiffness. To ensure connectivity, microstructures are mapped over an auxiliary field dictated by the optimized rotation variable. Finally, the analysis is carried out over mapped microstructures.Numerical experiments are performed for both 2D and 3D design domains to show the merit of the proposed parameterized microstructure. Various advantages along with existing challenges and future research will be discussed.

Topology optimization with design-dependent loading: An adaptive sensitivity-separation design variable update scheme

Yang Jiang1, Adeildo S. Ramos Jr.2, Glaucio H. Paulino1
1Georgia Institute of Technology; 2Federal University of Alagoas

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Compared with topology optimization with fixed loads only, the problem with design-dependent loading provides more realistic design, but also introduces additional difficulty including non-monotonous objective in the solution. To efficiently solve compliance minimization problems with design-dependent loads, we propose a new design variable update scheme. This update scheme relies on adaptive sensitivity separation to construct sub-problems with a non-monotonous approximation of the objective function. By iteratively solving these sub-problems using the Karush-Kuhn-Tucker conditions, the scheme updates the design variables efficiently. In addition to the standard formulation that limits the maximum material use, we propose a void formulation that limits the minimum material use, which provides infeasible design when self-weight is dominating over fixed loads. Reliability and efficiency of the update scheme is demonstrated by several numerical examples with both standard and void formulations. We observe that various optimal designs are obtained given different relative dominance of self-weight over fixed loads and different formulations.

Topology Optimization for Nonlinear Metamaterials Design with Bezier-based density representation algorithm

Hao Deng, Albert To
University of Pittsburgh

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Motivated by key advances in manufacturing techniques, the tailoring of materials to achieve novel properties such as stretchability or dissipation properties has been the focus of active research in engineering and materials science over the past decade. The goal of metamaterial design is to determine the optimal spatial layout to achieve a desired macroscopic constitutive response. However, the manufacturing abilities are the key factors to constrain the feasible design space, e.g. minimum length and geometry complexity. From the mathematical view, resolving above problems is equivalent to find an appropriate density field representation methodology to take the place of traditional density-based method. Traditional density-based method, where each element works as a variable, always results in complicated geometry with large number of small intricate features, while these small features are unfavorable for manufacture and lose its geometric accuracy after post-processing. To address above challenges, a new density field representation technique, named as Bezier-based explicit geometric representation scheme is proposed at the first time, where density field is described by Bezier-based Heaviside function. Bezier curves are widely used in computer graphics to produce curves which appear reasonably smooth at all scales. Compared to NURBS or B-spline, Bezier curves have less control parameters and easier to handle for sensitivities derivation, especially for distance sensitivities. Due to the powerful curve fitting ability, using Bezier curve to represent density field can explore design space effectively, and generate graceful structures without any intricate small features at borders. Furthermore, this density representation method is mesh independent and design variables are reduced significantly so that optimization problem can be solved efficiently using small-scale optimization algorithm, e.g. sequential quadratic programming. To demonstrate the powerful ability in design metamaterial of proposed density representation algorithm, two different metamaterial designs is presented, one design is to tailor material to present super stretchability under large deformation with fully recoverable properties. The other design is to optimize buckling-induced energy dissipation metamaterials to achieve desired energy dissipation capacity.

An XFEM Model Generation Library for Design Optimization

Keenan Doble1, Kurt Maute1, Miguel Aguiló2
1University of Colorado Boulder; 2Sandia National Laboratories

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An eXtended Finite Element Method (XFEM) computation model generation library is presented. The process to generate the XFEM model is introduced, namely, the mesh decomposition method, basis enrichment strategy and the creation of a geometry conforming pseudo-mesh. Some of the library's core features and the theory behind each feature will be discussed. The first core feature discussed is a method to decouple the XFEM model creation from the numerical analysis process. This decoupling affords a physics agnostic model and a minimally intrusive integration into existing finite element analysis packages. Integration into an existing finite element analysis package is demonstrated through the library's integration with Sandia National Laboratory's topology optimization package Plato Analyze. The second core capability discussed is the propogation of design sensitivities for use in levelset based design optimization problems. The library projects shape velocities defined within a geometry model onto the XFEM conformal pseudo-mesh. To this end, the design variables are abstracted such that they can be defined through the library's geometry interface or an external geometry engine. This capability will be demonstrated with a compliance minimization optimization problem. The final capability highlighted will be the library's ability to handle multiple geometries allowing for multi-material and multi-physics model generation. This will demonstrated with a composite fiber model.

Adaptive Level-Set Topology Optimization with the eXtended Finite Element Method and Hierarchical Mesh Refinement

Kurt Maute, Christian Messe, John Evans, Alireza Doostan
University of Colorado Boulder

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A broad range of topology optimization problems leads to designs with low volume fractions, i.e., the volume of the optimized design is much lower than the volume of the design domain. Using standard discretization approaches with uniformly refined meshes lead to impractical computational costs, in particular for 3D problems. Locally refining the mesh to resolve the geometry and the material layout as the design emerges in the optimization process has been identified a promising strategy and studied intensively in the past, mainly in the context of density methods. In this paper, we present a hierarchical mesh refinement scheme for level-set topology optimization. External boundaries and material interfaces are described by an explicit level-set method. The governing equations are discretized by eXtended finite element method. Starting with a coarse tensor mesh, hierarchical mesh refinement is a computationally efficient and robust method to generate 2D and 3D hexagonal and tetrahedral meshes. The refinement can be localized to regions of interest, such as in the vicinity of phase boundaries. Both, the level-set field and the state variables fields are discretized by hierarchically refined meshes, using the same or different interpolation orders. We will show with numerical examples that the proposed combination of methods leads to a computationally efficient approach to resolve local geometric features. Numerical examples include compliance and mass minimization problems with stress and eigen frequency constraints in 2D and 3D.

Concurrent optimization of structural topology and infill properties with a CBF-based level set method

Shikui Chen, Long Jiang, Yang Guo, Xianfeng David Gu
State University of New York at Stony Brook

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In this talk, the speaker will introduce a parametric level set method based on Cardinal basis functions to concurrently optimize the structural topology at the macroscale and the effective infill properties at the meso scale. The concurrent optimization is achieved by a computational framework combining a new parametric level set approach with mathematical programming. Within the proposed framework, both the structural boundary evolution and the effective infill property optimization can be driven by mathematical programming, which is more advantageous compared with the conventional PDE-driven level set approach. Moreover,the proposed approach will be more efficient in handling nonlinear problems with multiple constraints. Instead of using radial basis functions (RBF), in this paper, we propose to construct a new type of cardinal basis functions (CBF) for the level set function parameterization. The proposed CBF parameterization ensures an explicit impose of the lower and upper bounds of the design variables. This overcomes the intrinsic disadvantage of the conventional RBF-based parametric level set method, where the lower and upper bounds of the design variables oftentimes have to be set by trial and error. A variational distance regularization method is utilized in this research to regularize the level set function to be a desired distance-regularized shape. With the distance information embedded in the level set model, the wrapping boundary layer and the interior infill region can be naturally defined. The isotropic infill achieved via the mesoscale topology optimization is conformally fit into the wrapping boundary layer using the shape-preserving conformal mapping method, which leads to a hierarchical physical structure with optimized overall topology and effective infill properties. The proposed method is expected to provide a timely solution to the increasing demand for multiscale and multifunctional structure design.

A regularization scheme for explicit level-set XFEM topology optimization

Jorge Luis Barrera Cruz, Markus Geiss, Kurt Maute
University of Colorado Boulder

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Regularization of the level-set (LS) field is critical for both implicit and explicit LS-based topology optimization approaches. This is because locally too flat or steep LS functions negatively affect stability and rate of convergence of an optimization problem. In turn, a level-set function (LSF) with a uniform spatial gradient along phase boundaries greatly improves those properties. Traditionally, this is achieved by advancing the LS field through the solution of a Hamilton-Jacobi equation combined with a reinitialization scheme. This approach, however, may limit the maximum step size and introduce discontinuities in the design process. Alternatively, energy functionals have been proposed to control the gradient of the LSF. However, formulating well-posed optimization problems with these functionals can be challenging, and they require problem-specific tuning. To overcome the abovementioned shortcomings, a novel signed-distance field (SDF) regularization scheme that builds on a computationally efficient approach to construct signed distance field, namely the heat method, is introduced. To obtain the SDF, a transient heat conduction problem is solved first by applying zero initial conditions and enforcing a prescribed temperature at the interface.A distance field is then obtained as a solution of the Poisson's equation with the negative normalized temperature gradient as a volumetric heat source.Finally, the SDF with a unit norm gradient is obtained using the sign information of the design LSF, which is used as target LSF.To achieve continuous regularization of the design LSF, a penalty term is added to the objective function which minimizes the squared difference between the design and the target LSF as well as the difference in spatial gradients.The advantage of this approach is that a smooth and unique target field is used as reference for a locally enforced LS regularization. Using a differentiable penalty formulation to match the design with the target LSFs alleviates the need for reinitialization and circumvents introducing discontinuities in the optimization process. The implementation of this approach is straightforward within a finite element framework, and can easily be coupled with explicit LS methods.The governing equations of the proposed approach, as well as the ones describing the physical response of the system of interest, are discretized by the extended finite element method. Numerical examples for problems modeled by linear elasticity, nonlinear hyper-elasticity and the incompressible Navier-Stokes equations in two and three dimensions are presented to show the applicability of the proposed scheme to a broad range of design optimization problems.

A Novel Multigrid Preconditioning Scheme for Level-Set Topology Optimization on Hierarchically Refined B-Spline Meshes

Mathias Schmidt, John Evans, Kurt Maute, Christian Messe, Alireza Doostan
University of Colorado Boulder

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Topology Optimization benefits from adaptively refined meshes. The required resolution of the mesh is often not uniform. In areas with small geometric features and high field gradients a refined mesh is beneficial. In this work, hierarchically refined meshes using B-spline bases are used. In comparison with standard Lagrange bases, B-spline-based discretizations need less degrees of freedom for achieving the same accuracy and are known to generate better conditioned discretized systems.

In this study we consider a level-set topology optimization approach where the governing equations are discretized by the eXtended Finite Element Method. One of the major challenges of the XFEM in the context of topology optimization is the emergence of poorly conditioned linear systems. The issue is caused by particular intersection configurations where the contribution of a degree of freedom to the interpolation of a state variable field is much smaller than the ones of other degrees of freedom. The poor condition number hampers the convergence of the linear solution of the forward and adjoint problems by iterative solvers. Most standard preconditioners, such as incomplete LU approaches, are not well suited to efficiently treat this ill-conditioning issue. In his presentation, we introduce a novel geometric multigrid preconditioner that utilizes the hierarchical structure of an adaptively refined B-spline mesh.

Numerical examples show that the proposed preconditioning strategy is robust and insensitive to both, the mesh size and the intersection configuration. Moreover, the results show that the required number of iterations decrease as more refinement levels are used. These results indicate the promise of our new preconditioning strategy in the context of level set XFEM topology optimization.

Multiscale and Multiphysics Design Optimization (M2DO)

H. Alicia Kim
University of California San Diego

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This presentation will present the recent advances in level set topology optimization in challenging problems such as stress, buckling and nonlinear mechanics as well as design dependent loads such as thermal and pressure loads. With the design dependent loading appropriately considered in the calculation of shape sensitivity, topology optimization not only optimizes the structure for the applied loading, it also manipulates the design to reduce the design dependent loading thereby, achieving substantially greater savings. Numerical studies will be presented to demonstrate this clearly. With the advances of additive manufacturing recognizing its ability to manufacture feature sizes across several orders of magnitude without a substantially increased cost, a multiscale topology optimization formulation was introduced to enable coupling of two scale design using homogenization. The unique scientific contribution is the multiscale problem decomposition. The design requirements defined at the structural scale are decomposed into macroscale and microscale optimization problems and the constraints are formulated as an optimization problem to determine the optimum distribution of constraints such that the structural scale constraints are satisfied. This paves the way for determining the optimum materials specific for the application rather than typical approach of optimizing for the extremum properties. An alternative approach to addressing the multiscale optimization problem for additive manufacturing is to increase the mesh resolution and we introduce a novel computational algorithm for level set topology optimization that enables the possible mesh size by at least an order of magnitude greater than the traditional algorithm. These methods will demonstrate the state of the art capabilities of level set topology optimization.