A combined parametric shape optimization and ersatz material approach - Stingl, Michael; Wein, Fabian (Structural and Multidisciplinary Optimization, 2018)
This article describes a parametric shape optimization approach using vertical or horizontal structures with a fine parametrization of their center lines and profiles. In this context horizontal means a lateral connection from left to right and vertical means a bottom-up connection. These structures are projected to a pseudo density field associated with a fixed mesh using a differentiable mapping. This enables the use of existing topology optimization tools with respect to the solution of the state problem based on the pseudo density field. The approach belongs to a class of geometry projection onto a fictitious domain methods. It therefore shares the property that sensitivity analysis is reduced to extend the well known gradient calculation from topology optimization by chain using the sensitivity of the mapping from shape variables to pseudo density. The contribution lies in the combination with our specific shape parametrization and the associated regularization. Optimization problems can be formulated concurrently in terms of shape variables and pseudo density. We discuss regularization, periodicity constraints, symmetry formulations and overhang constraints in terms of shape variables. Volume and perimeter constraints are easily formulated in terms of the pseudo density. We see our approach as being particularly beneficial for certain problem classes where it may be difficult to restrict the design space, e.g. restricting isolated structures or holes or where a strict control of solid to void transition is necessary. Consequently, we show examples for phononic band gap maximization, boundary driven heat optimization and perimeter maximization for a flow problem. We also present a formulation of overhang constraints for additive manufacturing in terms of shape variables.
Keywords: Parametric shape optimization; Topology optimization

A nonlocal approach to graded surface modeling in topology optimization - Singh, Sukhminder; Pflug, Lukas; Wein, Fabian; Stingl, Michael (Structural and Multidisciplinary Optimization, 2025)
Additively manufactured structures often exhibit a correlation between their mechanical properties, such as stiffness, strength, and porosity, and their wall thickness. This correlation stems from the interplay between the manufacturing process and the properties of the filler material. In this study, we investigate the thickness-dependent effect on structural stiffness and propose a nonlocal integral model that introduces surface grading of Young’s modulus to capture this phenomenon. We incorporate this model into topology optimization for designing structures with optimized compliance subject to a volume constraint. Notably, elastically degraded surfaces penalize excessively thin features, effectively eliminating them from the optimized design. The effectiveness of the proposed framework is demonstrated through two- and three-dimensional examples.
CRIS-Team Scopus Importer:2025-05-16
Keywords: Additive manufacturing; Nonlocal integral model; Surface grading; Topology optimization

A pseudo density topology optimization approach in nonlinear electromagnetism applied to a 3D actuator - Seebacher, Philipp; Kaltenbacher, Manfred; Wein, Fabian; Lehmann, Henry (International Journal of Applied Electromagnetics and Mechanics, 2021)
A pseudo density topology optimization approach is introduced in nonlinear electromagnetism, which follows the SIMP (Solid Isotropic Material with Penalization) method. The adjoint approach is used to arrive at the sensitivity of the objective function. The elaborated optimization procedure includes nonlinearity of the ferromagnetic material. On the other hand, an applied linear characteristic of the ferromagnetic material results in a too small or wrong electromagnetic actuator dimension. Non-conforming grids are used to achieve a high accuracy by computing the discrete form. At the same time the number of finite elements decreases strongly. The implementation is applied to an electromagnetic actuator in 2D and 3D.
CRIS-Team WoS Importer:2021-04-09

A review on feature-mapping methods for structural optimization - Wein, Fabian; Dunning, Peter D.; Norato, Julián A. (Structural and Multidisciplinary Optimization, 2020)
In this review we identify a new category of methods for implementing and solving structural optimization problems that has emerged over the last 20 years, which we propose to call feature-mapping methods. The two defining aspects of these methods are that the design is parameterized by a high-level geometric description and that features are mapped onto a non-body-fitted mesh for analysis. One motivation for using these methods is to gain better control over the geometry to, for example, facilitate imposing direct constraints on geometric features, while avoiding issues with re-meshing. The review starts by providing some key definitions and then examines the ingredients that these methods use to map geometric features onto a fixed mesh. One of these ingredients corresponds to the mechanism for mapping the geometry of a single feature onto a fixed analysis grid, from which an ersatz material or an immersed-boundary approach is used for the analysis. For the former case, which we refer to as the pseudo-density approach, a test problem is formulated to investigate aspects of the material interpolation, boundary smoothing, and numerical integration. We also review other ingredients of feature-mapping techniques, including approaches for combining features (which are required to perform topology optimization) and methods for imposing a minimum separation distance among features. A literature review of feature-mapping methods is provided for shape optimization, combined feature/free-form optimization, and topology optimization. Finally, we discuss potential future research directions for feature-mapping methods.
CRIS-Team Scopus Importer:2020-08-07
Keywords: Feature-mapping; Fixed-grid; High-level design; Structural optimization

Automatic projection parameter increase for three-field density-based topology optimization - Dunning, Peter; Wein, Fabian (Structural and Multidisciplinary Optimization, 2025)
A method is proposed to automatically increase the threshold projection parameter in three-field density-based topology optimization to achieve near binary designs. The three-field method is composed of an element-wise design density field that is filtered and then passed through a smooth threshold projection function to compute the projected density field, which is then used to compute element properties, e.g., using a power law for stiffness. The sharpness of the threshold projection function is controlled by a parameter β. In this paper, a method is introduced to automatically increase this parameter during optimization by linking it to the change in objective function. Furthermore, the gray value indicator is added as a stopping criterion to guarantee the solution is near binary. This results in a method that does not need to be tuned for specific problems, or optimizers, and the same set of user-defined parameters can be used for a wide range of problems. However, a high value of the threshold projection parameter may cause convergence issues for some optimizers, such as the optimally criteria method, and an adaptive move limit strategy is introduced to overcome this problem. It is also shown that some problems require length-scale control to achieve a near binary design. The effectiveness of the method is demonstrated on several benchmark problems, including linear compliance, linear buckling, compliant mechanism, heat conduction, and geometrically nonlinear problems.
CRIS-Team Scopus Importer:2025-03-07
Keywords: Compliant mechanisms; Heat conduction; Linear buckling; Nonlinear geometry; Threshold projection; Topology optimization

Buckling optimization of additively manufactured cellular structures using numerical homogenization based on beam models - Hübner, Daniel; Herrnböck, Ludwig; Wein, Fabian; Mergheim, Julia; Steinmann, Paul; Stingl, Michael (Archive of Applied Mechanics, 2023)
Interest in components with detailed structures increased with the progress in advanced manufacturing techniques. Parts with lattice elements can provide improved global buckling stability compared to solid structures of the same weight. However, thin features are prone to local buckling. We present a two-scale optimization approach that simultaneously improves the local and global stability of parametrized graded lattice structures. Elastic properties and local buckling behavior are upscaled via homogenization based on geometric exact beam theory. To reduce computational effort, we construct a worst-case model for the homogenized buckling load factor, which acts as a safeguard against local buckling. We briefly discuss advantages and limitations by means of numerical examples.
CRIS-Team Scopus Importer:2023-11-03
Keywords: Beam theory; Buckling; Numerical homogenization; Two-scale optimization

Design and Additive Manufacturing of 3D Phononic Band Gap Structures Based on Gradient Based Optimization - Wormser, Maximilian; Wein, Fabian; Stingl, Michael; Körner, Carolin (Materials, 2017)
We present a novel approach for gradient based maximization of phononic band gaps. The approach is a geometry projection method combining parametric shape optimization with density based topology optimization. By this approach, we obtain, in a two dimension setting, cellular structures exhibiting relative and normalized band gaps of more than 8 and 1.6, respectively. The controlling parameter is the minimal strut size, which also corresponds with the obtained stiffness of the structure. The resulting design principle is manually interpreted into a three dimensional structure from which cellular metal samples are fabricated by selective electron beam melting. Frequency response diagrams experimentally verify the numerically determined phononic band gaps of the structures. The resulting structures have band gaps down to the audible frequency range, qualifying the structures for an application in noise isolation.
Keywords: phononic band gap; additive manufacturing; selective electron beam melting; cellular materials; metamaterials; topology optimization; parametric shape optimization; gradient based optimization

Design of Auxetic Structures via Mathematical Optimization - Schwerdtfeger, Jan; Wein, Fabian; Leugering, Günter; Singer, Robert; Körner, Carolin; Stingl, Michael; Schury, F. (Advanced Materials, 2011)
The optimization and manufacturing of an auxetic structure is presented. An inverse homogenization method is used to obtain the optimized geometry shown in the figure. The resulting structure is then produced using selective electron beam melting. The numerically predicted properties are experimentally verifie

Efficient spline design via feature-mapping for continuous fiber-reinforced structures - Greifenstein, Jannis; Letournel, Eloise; Stingl, Michael; Wein, Fabian (Structural and Multidisciplinary Optimization, 2023)
We present a feature-mapping topology optimization approach, in which curved features are parametrized as piecewise linear splines smoothly rounded by arcs. The motivation for our contribution to the tool set of feature-mapping methods is the optimization of structures manufactured by variable angle continuous fiber-reinforced filaments. For this reason, the feature's geometry should be able to represent long, curved fiber objects satisfying manufacturing constraints, such as minimum turning radius. The proposed model has been chosen with special care for rigorous continuous differentiability, as well as an efficient analytical evaluation of the signed distance field to the spline. The geometrical description and sensitivity analysis of the spline model are developed fully analytically and then mapped to a discretized pseudo-density field for finite element analysis. For the fiber-reinforced material formulation, we also present a new combine step for individual features, in which the best possible angle for the combined features is searched. The model and results are presented in a two-dimensional setting.
CRIS-Team WoS Importer:2023-05-12

Efficient two-scale optimization of manufacturable graded structures - Schury, Fabian; Stingl, Michael; Wein, Fabian (SIAM Journal on Scientific Computing, 2012)
A new method for the optimal design of inhomogeneous meso-structured components is presented. The method is based on a variant of the free material optimization (FMO) method. Instead of coupling a macroscopic model directly with the meso-scale in the framework of a two-scale approach, information from the meso-scale is incorporated into the FMO model by additional constraints. The FMO result, given in terms of optimized tensors, is interpreted by the inverse homogenization approach. Rather than only computing the meso-structure for each "FMO cell" independently, approximate continuity of the meso-structure is enforced. Thus, the final result is a manufacturable structurally graded material. The feasibility of the whole method is demonstrated by means of an academic example.
Keywords: free material optimization; inverse homogenization; structurally graded materials; topology optimization; numerical validation

Geometry: The leading parameter for the Poisson's ratio of bending-dominated cellular solids - Mitschke, Holger; Schury, Fabian; Mecke, Klaus; Wein, Fabian; Stingl, Michael; Schroder-Turk, Gerd E. (International Journal of Solids and Structures, 2016)
Control over the deformation behaviour that a cellular structure shows in response to imposed external forces is a requirement for the effective design of mechanical metamaterials, in particular those with negative Poisson's ratio. This article sheds light on the old question of the relationship between geometric microstructure and mechanical response, by comparison of the deformation properties of bar-and-joint frameworks with those of their realisation as a cellular solid made from linear-elastic material. For ordered planar tessellation models, we find a classification in terms of the number of degrees of freedom of the framework model: first, in cases where the geometry uniquely prescribes a single deformation mode of the framework model, the mechanical deformation and Poisson's ratio of the linearly-elastic cellular solid closely follow those of the unique deformation mode; the result is a bending-dominated deformation with negligible dependence of the effective Poisson's ratio on the underlying material's Poisson's ratio and small values of the effective Young's modulus. Second, in the case of rigid structures or when geometric degeneracy prevents the bending-dominated deformation mode, the effective Poisson's ratio is material-dependent and the Young's modulus (E) over tilde (cs) large. All analysed structures of this type have positive values of the Poisson's ratio and large values of (E) over tilde (cs). Third, in the case, where the framework has multiple deformation modes, geometry alone does not suffice to determine the mechanical deformation. These results clarify the relationship between mechanical properties of a linear-elastic cellular solid and its corresponding bar-and-joint framework abstraction. They also raise the question if, in essence, auxetic behaviour is restricted to the geometry-guided class of bending-dominated structures corresponding to unique mechanisms, with inherently low values of the Young's modulus. (C) 2016 Elsevier Ltd. All rights reserved.
Keywords: Cellular structures; Homogenisation; Floppy frameworks; Auxeticity; Mechanical metamaterial

Mechanical characterisation of a periodic auxetic structure produced by SEBM - Schwerdtfeger, Jan; Schury, Fabian; Stingl, Michael; Wein, Fabian; Singer, Robert; Körner, Carolin (physica status solidi (b), 2012)
We present a thorough investigation of the mechanical behaviour of a non-stochastic cellular auxetic structure. A combination of experimental and numerical methods is used to gain a deeper understanding of the mechanical behaviour and its dependence on the geometric properties of the cellular structure. The experimental samples are built from Ti-6Al-4V using selective electron beam melting, an additive manufacturing process giving the possibility to vary the geometry of the structure in a highly controlled manner. The use of finite element simulations and mathematical homogenisation allows us also to investigate off-axis properties of the cellular material. This leads to a more comprehensive understanding of the mechanical behaviour of the auxetics. Ultimately, the gained knowledge can be used to tailor auxetic materials to specific applications.
Keywords: auxetics; cellular materials; electron beam melting; finite element method; metals

On the effect of self-penalization of piezoelectric composites in topology optimization - Wein, Fabian; Kaltenbacher, Manfred; Kaltenbacher, Barbara; Leugering, Günter; Bänsch, Eberhard; Schury, Fabian (Structural and Multidisciplinary Optimization, 2011)
We investigate the occurrence of self-penalization in topology optimization problems for piezoceramicmechanical composites. Our main goal is to give physical interpretations for this phenomenon, i.e., to study the question why for various problems intermediate material values are not optimal in the absence of explicit penalization of the pseudo densities. In order to investigate this effect numerical experiments for several static and/or dynamic actuator and sensor objective functions are performed and their respective results are compared. The objective functions are mean transduction, displacement, sound power, electric potential, electric energy, energy conversion and electric power.

Slope constrained material design - Schury, Fabian; Stingl, Michael; Wein, Fabian (Structural and Multidisciplinary Optimization, 2012)
We investigate a class of constrained inverse homogenization problems. The complexity of the topological solution is restricted using slope constraint regularization. We show existence of the solution for the inverse optimization problem in function space and outline a converging approximation scheme. We demonstrate how a proper numerical implementation can lead to a stable material design approach. We finally describe results for a comprehensive set of numerical test cases.
Keywords: Slope constraints; Inverse homogenization; Topology optimization; SIMP

Topology optimization of a cantilevered piezoelectric energy harvester using stress norm constraints - Wein, Fabian; Kaltenbacher, Manfred; Stingl, Michael (Structural and Multidisciplinary Optimization, 2013)
Vibrational piezoelectric energy harvesters are devices which convert ambient vibrational energy into electric energy. Here we focus on the common cantilever type in which an elastic beam is sandwiched between two piezoelectric plates. In order to maximize the electric power for a given sinusoidal vibrational excitation, we perform topology optimization of the elastic beam and tip mass by means of the SIMP approach, leaving the piezoelectric plates solid. We are interested in the first and especially second resonance mode. Homogenizing the piezoelectric strain distribution is a common indirect approach increasing the electric performance. The large design space of the topology optimization approach and the linear physical model also allows the maximization of electric performance by maximizing peak bending, resulting in practically infeasible designs. To avoid such problems, we formulate dynamic piezoelectric stress constraints. The obtained result is based on a mechanism which differs significantly from the common designs reported in literature.
Keywords: Topology optimization; Energy harvester; Stress constraints

Topology optimization of a piezoelectric-mechanical actuator with single- and multiple-frequency excitation - Wein, Fabian; Kaltenbacher, Manfred; Leugering, Günter; Bänsch, Eberhard; Schury, Fabian (International Journal of Applied Electromagnetics and Mechanics, 2009)
We present the topology optimization of an assembly consisting of a piezoelectric layer attached to a plate with support. The optimization domain is the piezoelectric layer. Using the SIMP (Solid Isotropic Material with Penalization) method with forced vibrations by harmonic electrical excitation, we achieve a maximization of the dynamic displacement. We show that the considered objective function can be used under certain boundary conditions to optimize the sound radiation. The vibrational patterns resulting from the optimization are analysed in comparison with the modes from an eigenvalue analysis. Multiple-frequency optimization is achieved by adaptive weighted sums. As a second optimization criterion, a flat frequency response is integrated in the optimization process.
Keywords: Topology optimization; piezoelectric coupling; sound radiation

Topology optimization of smart piezoelectric transducers - Wein, Fabian (Friedrich-Alexander-Universität Erlangen-Nürnberg, 2011)
Numerical topology optimization based on the ersatz material model is very attractive in the research community and industry. Large scale nonlinear problems can be solved efficiently through the availability of appropriate optimizers, often resulting in non-intuitive solutions. However, topology optimization has not yet been established in the design of practical sensors and actuators. To this end we perform a thorough analysis and discussion of two exemplary piezoelectric devices, a single-frequency loudspeaker and a cantilevered energy harvester. With respect to the loudspeaker a broad range of objective functions is compared and discussed, culminating in a fully coupled piezoelectric-mechanical-acoustic near field topology optimization problem. Piezoelectric strain cancellation and acoustic short circuits need to be balanced with structural resonance in order to obtain close to resonance performance for almost arbitrary target frequencies. Providing appropriate initial designs proved to be essential for robust optimization. Cantilevered piezoelectric energy harvesters have been subject to various optimization approaches. However these have generally been based on reduced model assumptions. We present topology optimization of a realistic cantilevered energy harvester model. It proved to be necessary to use advanced topology optimization techniques, stress constraints to enforce practically feasible designs and Heaviside filtering for void features size control and for obtaining a black and white design pattern. To the best of our knowledge, this is the first time that dynamic piezoelectric stress constraints have been formulated for topology optimization. The obtained result is mechanism-based and interpretable to manufacture. This appears to be a novel finding in the field of cantilevered piezoelectric energy harvesting design. Performing numerical experiments, we were surprised to observe pronounced piezoelectric self-penalization, which means optimal black and white solutions without penalizing design interpolation and additional constraints beside box constraints on the design variable. This phenomenon is only rarely and briefly described in the literature. Within this thesis we perform initial heuristic steps in the analysis of the self-penalization phenomenon, which indeed appears in many different topology optimization problems. Once self-penalization is rigorously understood, our vision is to find methods supporting the self-penalizing effect and to obtain solutions potentially closer to the original problem than constrained and penalized ersatz problems. To this end we present oscillation constraints, a feature size control with independent solid and void feature size without enforcing intermediate pseudo material. <br
Keywords: topology optimization; piezoelectricity

Topology optimization of unsaturated flows in multi-material porous media: Application to a simple diaper model - Wein, Fabian; Chen, Nan; Iqbal, Naveed; Stingl, Michael; Avila, Marc (Communications in Nonlinear Science and Numerical Simulation, 2019)
We present a mathematical approach to optimize the material distribution for fluid transport in unsaturated porous media. Our benchmark problem is a simplified diaper model as an exemplary liquid absorber. Our model has up to three materials with vastly different properties, which in the reference configuration are arranged in parallel layers. Here, swelling is neglected and the geometry of a swollen diaper is assumed and treated as a porous medium of high porosity. The imbibition process is then simulated by solving an unsaturated flow problem based on Richards’ equation. Our aim is to maximize the amount of absorbed liquid by redistributing the materials. To this end, a density based multi-material topology optimization (based on the well known SIMP model) is performed. A sensitivity analysis for the nonlinear transient problem is provided, which enables the application of first order optimization solvers. We perform two- and three-material optimization and discuss several variants of the problem setting. We present designs with up to 45% more absorbed liquid compared to the reference configuration.
CRIS-Team Scopus Importer:2019-06-28
Keywords: Imbibition; Optimization; Permeability; Porous media

Two-Scale Optimization and Generation of Anisotropic Cellular Designs in the Context of Additive Manufacturing - Vu, Bich Ngoc; Wein, Fabian; Stingl, Michael (Computer-Aided Design, 2021)
Within the context of gradient-based optimization of cellular structures, we aim to optimize designs in a two-scale setting and generate respective geometric descriptions that are additively manufacturable in powder-bed systems. We analyze representative unit cells for their suitability and present a workflow to generate cellular designs by means of a parameterized unit cell, for which an implicit surface description is described. After performing homogenization-based optimization, we compare the predicted performance of the homogenized model to a full-resolution finite element (FE) analysis. Throughout the whole process, special attention is paid to the consistent integration of manufacturing limitations in terms of horizontal overhang and feature size. We discuss the performance of cellular components with maximized stiffness. Our numerical experiment for a single-load scenario demonstrates that using isotropic micro-structures is not beneficial, even when graded. We further observe that using anisotropic micro-structures leads to improvements only in very specific cases, e.g. with a uniformly distributed load. However, the conclusion may be different for anisotropic cells with non-uniform parameterization and other loading scenarios. Instead of conducting further tedious investigations for stiffness maximization, we finally demonstrate the benefit of cellular designs for a thermo-mechanical example.
CRIS-Team Scopus Importer:2021-07-16
Keywords: Cellular structures; Powder-based additive manufacturing; Structural optimization

Two-scale optimization of graded lattice structures respecting buckling on micro- and macroscale - Hübner, Daniel; Wein, Fabian; Stingl, Michael (Structural and Multidisciplinary Optimization, 2023)
Interest in components with detailed structures increased with the progress in advanced manufacturing techniques in recent years. Parts with graded lattice elements can provide interesting mechanical, thermal, and acoustic properties compared to parts where only coarse features are included. One of these improvements is better global buckling resistance of the component. However, thin features are prone to local buckling. Normally, analyses with high-computational effort are conducted on high-resolution finite element meshes to optimize parts with good global and local stability. Until recently, works focused only on either global or local buckling behavior. We use two-scale optimization based on asymptotic homogenization of elastic properties and local buckling behavior to reduce the effort of full-scale analyses. For this, we present an approach for concurrent local and global buckling optimization of parameterized graded lattice structures. It is based on a worst-case model for the homogenized buckling load factor, which acts as a safeguard against pure local buckling. Cross-modes residing on both scales are not detected. We support our theory with numerical examples and validations on dehomogenized designs, which show the capabilities of our method, and discuss the advantages and limitations of the worst-case model.
CRIS-Team Scopus Importer:2023-08-25
Keywords: Buckling; Cellular materials; Instability; Structural optimization; Two-scale

Two-scale optimization of graded lattice structures respecting buckling on micro- and macroscale - Hübner, Daniel; Wein, Fabian; Stingl, Michael (Structural and Multidisciplinary Optimization, 2023)
Interest in components with detailed structures increased with the progress in advanced manufacturing techniques in recent years. Parts with graded lattice elements can provide interesting mechanical, thermal, and acoustic properties compared to parts where only coarse features are included. One of these improvements is better global buckling resistance of the component. However, thin features are prone to local buckling. Normally, analyses with high-computational effort are conducted on high-resolution finite element meshes to optimize parts with good global and local stability. Until recently, works focused only on either global or local buckling behavior. We use two-scale optimization based on asymptotic homogenization of elastic properties and local buckling behavior to reduce the effort of full-scale analyses. For this, we present an approach for concurrent local and global buckling optimization of parameterized graded lattice structures. It is based on a worst-case model for the homogenized buckling load factor, which acts as a safeguard against pure local buckling. Cross-modes residing on both scales are not detected. We support our theory with numerical examples and validations on dehomogenized designs, which show the capabilities of our method, and discuss the advantages and limitations of the worst-case model.<br