Till Örebro universitet

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

In this study, a novel non-cooperative two-player game for minimizing static (Player 1) and dynamic (Player 2) compliances is introduced, implemented, and demonstrated using a multi-scale topology optimization framework for triply periodic minimal surface (TPMS)-based lattice structures. Player 1 determines the optimal macro-layout by minimizing the static compliance based on a micro-layout provided by Player 2. Conversely, player 2 identifies the optimal micro-layout (grading of the TPMS-based lattice structure) by minimizing the dynamic compliance given a macro-layout from Player 1. The multi-scale topology optimization formulations are derived using two density variables in each finite element. The first variable is the standard density, which dictates whether the finite element is void or contains the graded lattice structure and is governed by the rational approximation of material properties (RAMP) model. The second density variable represents the local relative density of the TPMS-based lattice structure, determining the effective orthotropic elastic properties of the finite element. The multi-scale game is implemented for three-dimensional problems, and solved using a Gauss-Seidel algorithm with sequential linear programming. It is numerically demonstrated for several benchmarks that the proposed multi-scale game generates equilibrium designs with strong performance for both static and harmonic load cases, effectively avoiding resonance at harmonic load frequencies. Validation is achieved through modal analyses of finite element models of the optimal designs.

In this work, a framework for data-driven sizing and shaping of topology optimization (TO) concepts is developed, implemented and demonstrated. The density field from a solid isotropic material with penalization (SIMP)-based TO solution is converted to an implicit surface-based geometry (ISG) by using regularized radial basis function networks (RBFN) with Wendland’s compactly supported radial basis functions. Sizing of the ISG is done locally by morphing operations and shaping is performed by applying free form deformations (FFD) on the stl-mesh which is generated from the ISG by a marching cube algorithm. The smooth FFD-based shaping is represented as a RBFN with cubic splines using a set of control points with corresponding prescribed deformations. Data-driven sizing and shaping of TO concepts are then performed by using multifidelity non-linear computer experiments and surrogate model-based design optimization. The developed and implemented framework is demonstrated for the well-known Messerschmitt-Bölkow-Blohm (MBB)-beam as well as an application of a flywheel to a compactor machine.

The use of lattice structures in design for additive manufacturing has quickly emerged as a popular and efficient design alternative for creating innovative multifunctional lightweight solutions. In particular, the family of triply periodic minimal surfaces (TPMS) studied in detail by Schoen for generating frame-or shell-based lattice structures seems extra promising. In this paper a multi-scale topology optimization approach for optimal macro-layout and local grading of TPMS-based lattice structures is presented. The approach is formulated using two different density fields, one for identifying the macro-layout and another one for setting the local grading of the TPMS-based lattice. The macro density variable is governed by the standard SIMP formulation, but the local one defines the orthotropic elasticity of the element following material interpolation laws derived by numerical homogenization. Such laws are derived for frame- and shell-based Gyroid, G-prime and Schwarz-D lattices using transversely isotropic elasticity for the bulk material. A nice feature of the approach is that the lower and upper additive manufacturing limits on the local density of the TMPS-based lattices are included properly. The performance of the approach is excellent, and this is demonstrated by solving several three-dimensional benchmark problems, e.g., the optimal macro-layout and local grading of Schwarz-D lattice for the established GE-bracket is identified using the presented approach.

In this article, five different formulations for establishing optimal ensembles of metamodels are presented and compared. The comparison is done by minimizing different norms of the residual vector of the leave-one-out cross-validation errors for linear, affine and convex combinations of 10 metamodels. The norms are taken to be the taxicab, the Euclidean and the infinity norm, respectively. The ensemble of metamodels consists of quadratic regression, Kriging with linear or quadratic bias, radial basis function networks with a-priori linear or quadratic bias, radial basis function networks with a-posteriori linear or quadratic bias, polynomial chaos expansion, support vector regression and least squares support vector regression. Eight benchmark functions are studied as ‘black-boxes’ using Halton and Hammersley samplings. The optimal ensembles are established for either one of the samplings and then the corresponding root mean square errors are established using the other sampling and vice versa. In total, 80 different test cases (5 formulations, 8 benchmarks and 2 samplings) are studied and presented. In addition, an established design optimization problem is solved using affine and convex combinations. It is concluded that minimization of the taxicab or Euclidean norm of the residual vector of the leave-one-out cross-validation errors for convex combinations of metamodels produces the best ensemble of metamodels. 

In this work, a topology optimization (TO) based framework for functional grading of triply periodic minimal surfaces (TPMS) based lattice structures is developed, implemented and demonstrated. Material interpolation laws of the gyroid, G-prime and Schwarz-D surfaces are derived by numerical homogenization for transversely isotropic elasticity and are represented as convex combinations of solid isotropic material with penalization (SIMP) and rational approximation of material properties (RAMP) models. These convex combinations are implemented in the TO-based compliance problem with new upper and lower bounds on the density variables representing the volume fraction limits of the lattices. The lower bound on the density variables is treated by introducing a sigmoid filter in the optimization loop forcing densities below the lower boundary towards zero. The optimal density solution is represented by Shepard interpolations or radial basis function networks, which, in turn, are utilized for the thickness grading of the TPMS-based lattices. In addition, the global boundary of the lattice structure is identified by support vector machines. Finally, a standard triangle language (STL) file is generated from the implicit surfaces by using marching cubes, which is utilized for further studies by nonlinear finite element analysis and to set up 3D printing of the optimal component quickly. The framework is demonstrated for the established L-shaped benchmark and the well-known General Electric engine bracket.

In this work, generative design optimization and characterization of triple periodic lattice structures in AlSi10Mg are considered. Structures with Gyroid, Schwarz-D and G-prime lattices are designed optimally by utilizing a generative design optimization approach. The approach is based on topology optimization, support vector machines (SVM), radial basis function networks (RBFN), morphing operations, design of experiments and metamodels. Firstly, topology optimization solutions are generated which are represented using SVM, secondly, sizing solutions obtained by setting the SIMP parameter equal to one are represented with RBFN. Thirdly, graded lattice structures using the RBFN are morphed together with the SVM to final conceptual designs. Fourthly, design of experiments of the conceptual designs are performed using non-linear finite element analyses (FEA) and, finally, metamodel-based design optimization is conducted using convex combinations of Kriging, RBFN, polynomial chaos expansion and support vector regression models. In order to validate the optimal designs, new tensile test specimens that include the periodic lattice structures are suggested. The specimens with all three lattices are manufactured in AlSi10Mg using direct metal laser sintering with an EOS M290 machine. Tensile tests of these specimens are then performed and validated using nonlinear FEA. The test specimens are also characterized with respect to geometry and defects by means of computed tomography, optical microscopy and scanning electron microscopy. The study demonstrates the high potential of using the proposed generative design optimization approach with triple periodic lattice structures for producing robust lightweight designs using additive manufacturing. In order to demonstrate the industrial relevance the established GE engine bracket is studied in the paper and discussed at the conference.

In this paper, we present a generative design optimization (GDO) approach for additive manufacturing (AM) by using topology optimization, support vector machines, cellular lattice structures (CLS), design of experiments, morphing and metamodel-based design optimization. By starting from appropriate design domains, a trade-off curve of design concepts is generated by SIMP-based topology optimization (TO). Then, a smooth implicit representation of the TO-solution is established by classifying the discrete density values using soft non-linear support vector machines (SVM). Instead of using the standard soft non-linear SVM of Cortez and Vapnik, we classify the TO solutions by using the 1-norm SVM of Mangasarian. In such manner, the classification is obtained by linear programming instead ofquadratic programming. The implicit SVM-model is further modified by incorporating cellular lattice structures, such as e.g. Gyroid lattice structures, by applying boolean operators. Design of experiments using finite element analysis are then set up by morphing the CLS-modified SVM models for different volume fractions. Finally, metamodel-based design optimization is performed by using optimal ensembles of polynomial regression models, Kriging, radial basis function networks, polynomial chaos expansion and support vector regression. The steps presented above constitute our proposed generative design optimization approach for additive manufacturing and are presented in more detail in the paper.

In this article, an approach for metamodel-based design optimization (MBDO) of topology optimization (TO) concepts is proposed by using support vector machines (SVMs) as geometric models of the concepts instead of traditional parametric computer aided design (CAD) models. In such a manner, an efficient approach for the MBDO-driven design of TO-based concepts is obtained. An implicit hypersurface representing the TO-based concept is generated by classifying the TO-solutions of zeros and ones by using the 1-norm SVM of Mangasarian. The implicit SVM-based hypersurfaces are then utilized to set up designs of experiments of nonlinear finite element analyses by morphing the TO-based concepts by using Boolean and blending operations. Finally, MBDO is performed by using an ensemble of metamodels consisting of quadratic regression, Kriging, radial basis function networks, polynomial chaos expansion and support vector regression models. The proposed MBDO framework is demonstrated by minimizing the mass of a three-dimensional design domain with a constraint on the plastic limit load. The performance of the approach is most promising.