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Strömberg, N. (2024). A new multi-scale topology optimization framework for optimal combinations of macro-layouts and local gradings of TPMS-based lattice structures. Mechanics based design of structures and machines, 52(1), 257-274
Open this publication in new window or tab >>A new multi-scale topology optimization framework for optimal combinations of macro-layouts and local gradings of TPMS-based lattice structures
2024 (English)In: Mechanics based design of structures and machines, ISSN 1539-7734, E-ISSN 1539-7742, Vol. 52, no 1, p. 257-274Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
Taylor & Francis Group, 2024
Keywords
Topology optimization, functional grading, triply periodic minimal surfaces, numerical homogenization, additive manufacturing constraints
National Category
Applied Mechanics
Identifiers
urn:nbn:se:oru:diva-100814 (URN)10.1080/15397734.2022.2107538 (DOI)000837639300001 ()2-s2.0-85135567445 (Scopus ID)
Funder
Vinnova
Available from: 2022-08-24 Created: 2022-08-24 Last updated: 2024-02-05Bibliographically approved
Strömberg, N. (2021). A Multi-scale Topology Optimization Approach for Optimal Macro-layout and Local Grading of TPMS-based Lattices. In: Proceedings of ASME 2021 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference (IDETC-CIE2021) (Volume 3): Volume 3A: 47th Design Automation Conference (DAC). Paper presented at ASME 2021 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, IDETC/CIE, (Virtual Conference), August 17–20, 2021. American Society of Mechanical Engineers
Open this publication in new window or tab >>A Multi-scale Topology Optimization Approach for Optimal Macro-layout and Local Grading of TPMS-based Lattices
2021 (English)In: Proceedings of ASME 2021 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference (IDETC-CIE2021) (Volume 3): Volume 3A: 47th Design Automation Conference (DAC), American Society of Mechanical Engineers , 2021Conference paper, Published paper (Refereed)
Abstract [en]

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.

Place, publisher, year, edition, pages
American Society of Mechanical Engineers, 2021
Series
International Design Engineering Technical Conferences & Computers and Information in Engineering Conference (IDETC/CIE)
National Category
Applied Mechanics
Identifiers
urn:nbn:se:oru:diva-97583 (URN)10.1115/DETC2021-67163 (DOI)2-s2.0-85119950674 (Scopus ID)9780791885383 (ISBN)
Conference
ASME 2021 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, IDETC/CIE, (Virtual Conference), August 17–20, 2021
Available from: 2022-02-18 Created: 2022-02-18 Last updated: 2022-08-31Bibliographically approved
Strömberg, N. (2021). Comparison of optimal linear, affine and convex combinations of metamodels. Engineering optimization (Print), 53(4), 702-718
Open this publication in new window or tab >>Comparison of optimal linear, affine and convex combinations of metamodels
2021 (English)In: Engineering optimization (Print), ISSN 0305-215X, E-ISSN 1029-0273, Vol. 53, no 4, p. 702-718Article in journal (Refereed) Published
Abstract [en]

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. 

Place, publisher, year, edition, pages
Taylor & Francis, 2021
Keywords
Convex combination, ensemble, Metamodel
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:oru:diva-81874 (URN)10.1080/0305215X.2020.1746781 (DOI)000532162600001 ()2-s2.0-85084258455 (Scopus ID)
Available from: 2020-05-19 Created: 2020-05-19 Last updated: 2021-08-31Bibliographically approved
Strömberg, N. (2021). Optimal grading of TPMS-based lattice structures with transversely isotropic elastic bulk properties. Engineering optimization (Print), 53(11), 1871-1883
Open this publication in new window or tab >>Optimal grading of TPMS-based lattice structures with transversely isotropic elastic bulk properties
2021 (English)In: Engineering optimization (Print), ISSN 0305-215X, E-ISSN 1029-0273, Vol. 53, no 11, p. 1871-1883Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
Taylor & Francis, 2021
Keywords
Triply periodic minimal surfaces, optimal grading, homogenization, topology optimization, STL-files
National Category
Mechanical Engineering
Identifiers
urn:nbn:se:oru:diva-87636 (URN)10.1080/0305215X.2020.1837790 (DOI)000588183400001 ()2-s2.0-85096148617 (Scopus ID)
Funder
Vinnova
Available from: 2020-11-27 Created: 2020-11-27 Last updated: 2021-12-08Bibliographically approved
Karlsson, P., Pejryd, L. & Strömberg, N. (2020). Generative Design Optimization and Characterization of Triple Periodic Lattice Structures in AlSi10Mg. In: Mirko Meboldt; Christoph Klahn (Ed.), Industrializing Additive Manufacturing: Proceedings of AMPA2020. Paper presented at 2nd International Conference on Additive Manufacturing for Products and Applications (AMPA 2020), Zürich, Switzerland, September 1-3, 2020 (pp. 3-16). Cham: Springer
Open this publication in new window or tab >>Generative Design Optimization and Characterization of Triple Periodic Lattice Structures in AlSi10Mg
2020 (English)In: Industrializing Additive Manufacturing: Proceedings of AMPA2020 / [ed] Mirko Meboldt; Christoph Klahn, Cham: Springer, 2020, p. 3-16Conference paper, Published paper (Refereed)
Abstract [en]

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.

Place, publisher, year, edition, pages
Cham: Springer, 2020
Keywords
Generative design, Lattice structures, AlSi10Mg
National Category
Mechanical Engineering Materials Engineering Metallurgy and Metallic Materials Applied Mechanics
Research subject
Mechanical Engineering
Identifiers
urn:nbn:se:oru:diva-85697 (URN)10.1007/978-3-030-54334-1_1 (DOI)9783030543334 (ISBN)9783030543341 (ISBN)
Conference
2nd International Conference on Additive Manufacturing for Products and Applications (AMPA 2020), Zürich, Switzerland, September 1-3, 2020
Available from: 2020-09-12 Created: 2020-09-12 Last updated: 2023-03-02Bibliographically approved
Strömberg, N. (2019). A Generative Design Optimization Approach for Additive Manufacturing. In: F. Auricchio, E. Rank, P. Steinmann, S. Kollmannsberger and S. Morganti (Ed.), Second International Conference on Simulation for Additive Manufacturing: . Paper presented at 2nd International Conference on Simulation for Additive Manufacturing (Sim-AM 2019), Pavie, Italy, September 11-13, 2019 (pp. 130-141). Barcelona, Spain: International Centre for Numerical Methods in Engineering (CIMNE)
Open this publication in new window or tab >>A Generative Design Optimization Approach for Additive Manufacturing
2019 (English)In: Second International Conference on Simulation for Additive Manufacturing / [ed] F. Auricchio, E. Rank, P. Steinmann, S. Kollmannsberger and S. Morganti, Barcelona, Spain: International Centre for Numerical Methods in Engineering (CIMNE) , 2019, p. 130-141Conference paper, Published paper (Refereed)
Abstract [en]

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.

Place, publisher, year, edition, pages
Barcelona, Spain: International Centre for Numerical Methods in Engineering (CIMNE), 2019
Keywords
Topology optimization, Support vector machines, Lattice Structures, Metamodels
National Category
Applied Mechanics
Identifiers
urn:nbn:se:oru:diva-80213 (URN)000563504100012 ()978-84-949194-8-0 (ISBN)
Conference
2nd International Conference on Simulation for Additive Manufacturing (Sim-AM 2019), Pavie, Italy, September 11-13, 2019
Available from: 2020-02-26 Created: 2020-02-26 Last updated: 2020-09-16Bibliographically approved
Strömberg, N. (2019). Efficient detailed design optimization of topology optimization concepts by using support vector machines and metamodels. Engineering optimization (Print)
Open this publication in new window or tab >>Efficient detailed design optimization of topology optimization concepts by using support vector machines and metamodels
2019 (English)In: Engineering optimization (Print), ISSN 0305-215X, E-ISSN 1029-0273Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
Taylor & Francis, 2019
Keywords
Support vector machines, topology optimization, metamodels
National Category
Applied Mechanics
Identifiers
urn:nbn:se:oru:diva-75953 (URN)10.1080/0305215X.2019.1646258 (DOI)000481217300001 ()
Available from: 2019-08-30 Created: 2019-08-30 Last updated: 2019-08-30Bibliographically approved
Strömberg, N. (2019). Reliability-based Design Optimization by using Ensemble of Metamodels. In: Proceedings of the 3rd International Conference on Uncertainty Quantification in Computational Sciences and Engineering: . Paper presented at 3rd International Conference on Uncertainty Quantification in Computational Sciences and Engineering (UNCECOMP 2019), Crete, Greece, June 24-26, 2019. ECCOMAS
Open this publication in new window or tab >>Reliability-based Design Optimization by using Ensemble of Metamodels
2019 (English)In: Proceedings of the 3rd International Conference on Uncertainty Quantification in Computational Sciences and Engineering, ECCOMAS , 2019Conference paper, Published paper (Refereed)
Place, publisher, year, edition, pages
ECCOMAS, 2019
National Category
Applied Mechanics
Identifiers
urn:nbn:se:oru:diva-80212 (URN)
Conference
3rd International Conference on Uncertainty Quantification in Computational Sciences and Engineering (UNCECOMP 2019), Crete, Greece, June 24-26, 2019
Available from: 2020-02-26 Created: 2020-02-26 Last updated: 2020-03-17Bibliographically approved
Strömberg, N. (2018). An Eulerian-based Thermo-Flexible Multi-Body Approach for Simulating Rig Testing of Disc Brakes. In: Luis Rodriguez-Tembleque; M.H. Ferri Aliabadi (Ed.), Advances in Computational Coupling and Contact Mechanics: (pp. 179-195). World Scientific
Open this publication in new window or tab >>An Eulerian-based Thermo-Flexible Multi-Body Approach for Simulating Rig Testing of Disc Brakes
2018 (English)In: Advances in Computational Coupling and Contact Mechanics / [ed] Luis Rodriguez-Tembleque; M.H. Ferri Aliabadi, World Scientific, 2018, p. 179-195Chapter in book (Refereed)
Abstract [en]

In this chapter, we propose an Eulerian-based thermo-flexible multi-body approach in order to simulate rig testing of disc brake systems accurately and efficiently. A multi-body model of the disc, an assembly of flywheels and the shaft connecting the disc and flywheels is coupled to an Eulerian-based thermo-mechanical finite element model of the disc-pad system. By utilizing the Eulerian framework of the disc, the contact interface is modelled most accurately with Signorini contact, Coulomb frictional heating with a new temperature-dependent friction model that includes fading at high temperatures and Archard wear with a temperature-dependent wear coefficient. The governing equations are treated with a sequential approach, where first the mechanical contact problem is solved using the augmented Lagrangian approach with a non-smooth Newton method. Then, the multi-body model is solved for the brake moment obtained from the mechanical contact analysis using the average acceleration method. Finally, heat balance for the system including the frictional power from the two previous steps is obtained with the trapezoidal rule and formulating the nonlinear equations as a system of linear equations. Here, the non-symmetric convection matrix is stabilized by adding artificial conduction according to the streamline-upwind approach. The proposed sequential approach is implemented in an in-house code and utilized to study a vented disc-pad system to a heavy truck. This is discussed at the end of the chapter, showing the development of heat bands, hot spots and corresponding residual stresses.

Place, publisher, year, edition, pages
World Scientific, 2018
Series
Computational and experimental methods in structures, ISSN 2044-9283 ; 11
Keywords
Contact, Thermomechanics, Heat bands, Disc brakes
National Category
Applied Mechanics
Research subject
Mechanical Engineering; Computer Engineering
Identifiers
urn:nbn:se:oru:diva-83835 (URN)10.1142/q0139 (DOI)9781786344779 (ISBN)9781786344793 (ISBN)
Available from: 2020-06-28 Created: 2020-06-28 Last updated: 2023-03-09Bibliographically approved
Strömberg, N. (2018). Automatic Postprocessing of Topology Optimization Solutions by Using Support Vector Machines. In: Proceedings of the ASME Design Engineering Technical Conference: Volume 2B. Paper presented at ASME International Design Engineering Technical Conferences (IDETC) / Computers and Information in Engineering Conference (CIE), Quebec, Canada, August 26–29, 2018. American Society of Mechanical Engineers (ASME)
Open this publication in new window or tab >>Automatic Postprocessing of Topology Optimization Solutions by Using Support Vector Machines
2018 (English)In: Proceedings of the ASME Design Engineering Technical Conference: Volume 2B, American Society of Mechanical Engineers (ASME) , 2018Conference paper, Published paper (Refereed)
Abstract [en]

The postprocessing step from the density result in topology optimization to a parametric CAD model is typically most time consuming and usually involves several hands on maneuvers by an engineer. In this paper we propose an approach in order to automate this step by using soft non-linear support vector machines (SVM). Our idea is to generate the boundaries separating regions of material (elements with densities equal to one) and no material (elements with densities equal zero) obtained from topology optimization automatically by using SVM. The hypersurface of the SVM can then in the long run be explicitly implemented in any CAD software. In this work we generate these hypersurfaces by solving the dual formulation of the SVM with soft penalization and nonlinear kernel functions using quadratic programming or the sequential minimal optimization approach. The proposed SVM-based postprocessing approach is studied on topology optimization results of orthotropic elastic design domains with mortar contact conditions studied most recently in a previous work. The potential energy of several bodies with non matching meshes is maximized. In such manner no extra adjoint equation is needed. Intermediate density values are penalized using SIMP or RAMP, and the regularization is obtained by applying sensitivity or density filters following the approaches of Sigmund and Bourdin. The study demonstrates that the SVM-based postprocessing approach automatically generates proper hypersurfaces which can be used efficiently in the CAD modelling.

Place, publisher, year, edition, pages
American Society of Mechanical Engineers (ASME), 2018
National Category
Applied Mechanics
Identifiers
urn:nbn:se:oru:diva-73412 (URN)10.1115/DETC2018-85051 (DOI)000461130700001 ()2-s2.0-85057014507 (Scopus ID)
Conference
ASME International Design Engineering Technical Conferences (IDETC) / Computers and Information in Engineering Conference (CIE), Quebec, Canada, August 26–29, 2018
Funder
Vinnova
Available from: 2019-03-29 Created: 2019-03-29 Last updated: 2019-03-29Bibliographically approved
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