Conference Agenda

We live in a world of exploding complexity with enormous challenges. Digital twins, tightly integrating the real and the digital world, are a key enabler for decision making in the context of complex systems. While the digital twin has become an intrinsic part of the product creation process, its true power lies in the connectivity of the digital representation with its physical counterpart.

To be able to use a digital twin scalable in this context, the concept of an executable digital twin has been proposed. An executable digital twin is a stand-alone and self-contained executable model for a specific set of behaviors in a specific context. It can be leveraged by anyone at any point in lifecycle. To achieve this, a broad toolset of mathematical technologies is required - ranging from model order reduction, calibration to hybrid physics- and data-based models.

In this presentation, we review the concept of executable digital twins, address mathematical key building blocks such as model order reduction, real-time models, state estimation, and co-simulation and detail its power along a few selected use cases.

The numerical simulation of nonlinear problems in contact mechanics and mixed-dimensional coupling poses significant challenges for high-performance computing (HPC). The considerable computational effort introduced by the evaluation of discrete contact or mixed-dimensional coupling operators requires an efficient framework that scales well on parallel hardware architectures and is, thus, suitable for the solution of high-fidelity models with potentially many million degrees of freedom. Another unsolved task is the efficient solution of the arising linear systems of equations on parallel computing clusters. In this study, we investigate the scalability of computational strategies and algebraic multigrid (AMG) solvers to address these challenges effectively and ultimately target a faster time-to-solution. Two problem classes of utmost practical relevance serve as prototypes for this endeavor.

First, we investigate the finite element analysis of nonlinear contact problems based on non-matching mortar interface discretizations. Mortar methods enable a variationally consistent imposition of almost arbitrarily complex coupling conditions but come with considerable computational effort for the evaluation of the discrete coupling operators (especially in 3D). We identify bottlenecks in parallel data layout and domain decomposition that hinder a truly efficient evaluation of the mortar operators on modern-day HPC systems and then propose computational strategies to restore optimal parallel communication and scalability. In particular, we suggest a dynamic load balancing strategy in combination with a geometrically motivated reduction of ghosting data. Using increasingly complex 3D examples, we demonstrate strong and weak scalability of the proposed algorithms up to 480 parallel processes. In addition to the computational strategies, we investigate the integration of tailored AMG preconditioners for the resulting saddle point-type linear systems of equations into a then fully scalable simulation pipeline.

Second, we present the mixed-dimensional interaction of slender fiber- or rod-like structures with surrounding solid volumes, thus leading to a 1D-3D approach that we refer to as beam-to-solid coupling. In terms of practical applications, natural and artificial fiber-reinforced materials (e.g., biological tissue, composites) come to mind. Again, not so different from contact problems, the main challenges on the way to a scalable computational framework lie in the efficient and parallelizable evaluation of the underlying mixed-dimensional coupling operators and the design of tailored AMG preconditioners. Here, we will focus on a novel physics-based block preconditioning approach based on AMG that uses multilevel ideas to approximate the block inverses appearing in the system. Eventually, we will assess the performance and the weak scalability of the proposed block preconditioner using large-scale numerical examples.

Reduced Order Models (ROMs) have been widely used to efficiently solve large-scale problems in many fields including computational fluid dynamics (CFD) [1]. ROMs techniques allow to replace the expansive Full Order Model (FOM), by a ROM that captures the essential features of the system while significantly reducing the computational cost. In this work, we draw inspiration from [2] to implement a reduced basis (RB) method for model reduction of the Shallow Water Equations (SWEs) using Proper Orthogonal Decomposition (POD) and Artificial Neural Networks (ANNs). This method, referred to as POD-NN, starts with the POD technique to construct a reduced basis, and then makes us of an ANN to learn the associated coefficients in the reduced basis. It follows an offline-online strategy: the POD reduced basis along with the training of the ANN are performed in an offline stage, and then the surrogate model can be used for hyper-fast predictions. The process is non-intrusive since it does not require opening the black box of the FOM. The developed method is tested [3] on a real data set aiming at simulating an inundation of the Aude river (Southern France). The results show that the proposed method can achieve significant computational savings while maintaining satisfactory accuracy on the hydraulic variables of interest compared to the full-order hydraulic model. The proposed method is able to capture the key features of the SWEs in particular the wave propagation. Overall, the proposed non-intrusive POD-NN method offers a promising approach for ROM of SWEs while being affordable in view of fast real time inundation simulations.

[1] Benner, P., Schilders, W., Grivet-Talocia, S., Quarteroni, A., Rozza, G., & Miguel Silveira, L. (2020). Model Order Reduction: Volume 2: Snapshot-Based Methods and Algorithms (p. 348). De Gruyter.

[2] Hesthaven, J. S., & Ubbiali, S. (2018). Non-intrusive reduced order modeling of nonlinear problems using neural networks. Journal of Computational Physics, 363, 55-78.

[3] IMT-INSA, INRAE et al., “DassFlow: Data Assimilation for Free Surface Flows”, Open source computational software. https://www.math.univ-toulouse.fr/DassFlow

Let us consider wave propagation problems over 2-dimensional domains with piecewise-linear boundaries, possibly including scatterers. Under some assumptions on the initial conditions and forcing terms, we have proposed an approximation of the propagating wave as the sum of some special nonlinear space-time functions. Each term in this sum identifies a particular ray, modeling the result of a single reflection or diffraction effect. In this talk I will describe an algorithm for identifying such rays automatically, based on the domain geometry.

To showcase our proposed method, I will present several numerical examples, such as waves scattering off wedges and waves propagating through a room in presence of obstacles.

This contribution is motivated by the combined advantages of an integrated framework from CAD geometries to simulation in real time. In a typical workflow for design and shape optimization, multiple simulations are required for all possible designs represented by different geometrical parameters. This might entail a high computational cost in particular for real world, engineering applications. The development of efficient reduced order models that enable fast parametric analysis is essential for such applications. At the same time, the capabilities of splines and isogeometric analysis allow for flexible geometric design and higher-order continuity in the analysis. In CAD design, trimmed multi-patch geometries are widely used to represent complex shapes. The presence of geometric parameters introduces challenges for efficient reduced order modeling of problems formulated on such unfitted geometries. We propose a localized reduced basis method to circumvent the shortcomings of standard reduced order models in this context [1]. In this talk we present the developed strategy and address fast parametric analysis of problems in structural mechanics. The construction of efficient reduced order models for geometries described by multiple trimmed patches as well as their use in parametric shape optimization will be discussed. Numerical examples will be presented to demonstrate the accuracy and computational efficiency of the method.

[1] M. Chasapi, P. Antolin, A. Buffa, A localized reduced basis approach for unfitted domain methods on parameterized geometries, Comput. Methods Appl. Mech. Engrg. 410 (2023) 115997.

In many applications in Computer Aided Engineering, like parametric studies, structural optimization or virtual material design, a large number of almost similar models have to be simulated. Although the individual scenarios may differ only slightly in both space and time, the same effort is invested for every single new simulation with no account for experience and knowledge from previous simulations. Therefore, we have developed a method that combines data-based Model Order Reduction (MOR) and reanalysis, thus exploiting knowledge from previous simulation runs to accelerate computations in multi-query contexts. While MOR allows reducing model fidelity in space and time without significantly deteriorating accuracy, reanalysis uses results from previous computations as a predictor or preconditioner.

The workflow of our method, named Reduced Model Reanalysis (RMR), is divided into an offline and online phase. In the offline phase, data are generated to cover a wide range of the parameter space. From this data a surrogate model is learned in a reduced space using regression algorithms from the field of machine learning. Depending on the requirements of the system, different regression algorithms are favorable, e.g. linear regression, a k-nearest neighbor algorithm, a neural network, or a Gaussian process. The models are learned in the reduced space due to the prohibitively large number of degrees of freedom of the full finite element model. The reduced subspaces are obtained via a snapshot POD (proper orthogonal decomposition). In the online phase, approximations of all relevant solution quantities are obtained from the surrogate model. Their projection to the full space provides predictors that allow for an accelerated solution of the system in comparison to a standard structural mechanics computation.

In the case of nonlinear stability analysis this method can for example be used to accelerate the exact computation of critical points by the method of extended systems. Data generation in the offline phase is also accelerated by a newly developed adaptive time stepping scheme. With this scheme the number of steps to approach critical points with a path following scheme can be significantly reduced. Further potential fields of application of RMR are general nonlinear static and transient problems, with particular challenges as soon as path-dependence comes into play.

Multiscale methods are often employed to study the effect of microstructure on macroscopic behaviour. For non-linear problems, these usually result in a two-scale formulation, where macro- and microstructure are simultaneously solved and coupled. If the microstructural features are much smaller compared to the macrostructural size, its effective behavior can be sufficiently predicted with first-order computational homogenization. However, if scale separation cannot be assumed or non-local effects due to buckling, softening, etc., emerge, higher-order methods, such as second-order homogenization [1], need to be considered. This formulation contains the second gradient of the displacement field, giving rise to a length-scale associated with the length-scale of the underlying unit cell, thus making it possible to capture size and non-local effects. Solving such problems is currently computationally expensive and typically infeasible for realistic applications, which limits the applicability of this method.

In this work, we address this issue by developing a reduced order model for second-order computational homogenization scheme based on Proper Orthogonal Decomposition. We consider different numerical examples and discuss different training strategies, computational savings and accuracy of the surrogate model.

Acknowledgements: This result is part of a project that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 Research and Innovation Programme (Grant Agreement No. 818473).

[1] Kouznetsova, V., Geers, M.G.D. and Brekelmans, W.A.M. (2002), Multi-scale constitutive modelling of heterogeneous materials with a gradient-enhanced computational homogenization scheme. Int. J. Numer. Meth. Engng., 54: 1235-1260. https://doi.org/10.1002/nme.541

Manifold learning techniques such as Laplacian Eigenmaps (LE) [1] are commonly applied in fields like image and speech processing, to extract nonlinear trends from large sets of high-dimensional data [2]. Such techniques are also intriguing as model order reduction methods in multiscale solid mechanics: LE can capture nonlinearities in the solution manifolds of discretised physical problems [3]. Compared to POD-based algorithms, this may result in reduced order models yielding more accurate results with fewer parameters and lower computational effort [3]. Consequently, manifold learning techniques have been applied successfully to problems in fluid mechanics [3] and elastodynamics [4].

In the framework of the FE² method – in which computations on microscale representative volume elements (RVEs) are performed at each Gauss point of a macroscopic problem [5] – the payoff of such computational cost reduction may also be significant.

This contribution discusses the application of LE to model order reduction for RVE computations. Nonlinear, hyperelastic and elastoplastic behaviour is considered. The area of application comes with unique challenges and opportunities: for example, the mapping between reduced and original spaces and the projection of residuals onto reduced bases is not trivial [3]. On the other hand, the underlying PDEs [4] and the parametrisation of the RVE problem via a macroscopic deformation gradient and history variables [5] imply a strong (nonlinear) correlation in the unknown displacement degrees of freedom to be reduced. This talk will explore some of these challenges and opportunities.

[1] Belkin, Mikhail, and Partha Niyogi. "Laplacian eigenmaps for dimensionality reduction and data representation." Neural computation 15, no. 6 (2003): 1373-1396.

[2] Lee, John A., and Michel Verleysen. Nonlinear dimensionality reduction. Vol. 1. New York: Springer, 2007.

[3] Pyta, Lorenz Matthias. "Modellreduktion und optimale Regelung nichtlinearer Strömungsprozesse." PhD diss., Dissertation, RWTH Aachen University, 2018.

[4] Millán, Daniel, and Marino Arroyo. "Nonlinear manifold learning for model reduction in finite elastodynamics." Computer Methods in Applied Mechanics and Engineering 261 (2013): 118-131.

[5] Schröder, Jörg. "A numerical two-scale homogenization scheme: the FE 2-method." Plasticity and beyond: microstructures, crystal-plasticity and phase transitions (2014): 1-64.

Plasticity is the result of the motion and interaction of discrete dislocations in a crystalline material. Modelling plasticity at the crystal level based on discrete dislocation dynamics (DDD) is challenging due to the complexities associated with the dislocation activities of different slip planes. A data-driven approach provides an alternative method for simulating the complex behavior associated with plasticity at a small scale. We use methods based on dynamic mode decomposition1 (DMD) to analyze the DDD data2. We built reduced-order models for describing system dynamics with a few dominant modes. The models are built upon datasets of different physical resolution, e.g. dislocation density information resolved on the slip system level based on total dislocation densities, dislocation density vectors, or second-order dislocation alignment tensors. Different levels of spatial resolution are used to evaluate the effectiveness of models in reconstruction of the analysed data.

The modelling approach is then extended to forecast material response beyond the training dataset, for which we adopt more general (non-linear) Koopman operator theory and advanced stabilized DMD schemes, like shift invariant (physically informed) DMD or optimized DMD3. The different schemes are compared in their ability to predict the nonlinear behaviour in crystal plasticity from the DDD data.

REFERENCES

[1] Schmid, Peter J. "Dynamic mode decomposition of numerical and experimental data." Journal of fluid mechanics 656 (2010): 5-28.

[2] Akhondzadeh, Sh, Ryan B. Sills, Nicolas Bertin, and Wei Cai. "Dislocation density-based plasticity model from massive discrete dislocation dynamics database." Journal of the Mechanics and Physics of Solids 145 (2020): 104152.

[3] Askham, Travis, and J. Nathan Kutz. "Variable projection methods for an optimized dynamic mode decomposition." SIAM Journal on Applied Dynamical Systems 17, no. 1 (2018): 380-416.

The availability of high quality µ-CT images of materials allows for detailed multiscale simulation workflows in digital material characterization. In this case, data driven hybrid machine learning approaches are used to speed up full field approaches. Efficient and performance implementations of such data driven methods are essential for them being used for industrial applications.

This work concerns DMN (Deep Material Network) whose potential applications were exploited recently [1,2,3]. They only need linear elastic training data to identify equivalent laminate microstructure, which can be used to predict nonlinear behavior.

The industrial applicability of the DMN for short fiber reinforced plastic is investigated by comparing its speed and accuracy against direct numerical simulation results[4,5] on RVEs[6] using different physically nonlinear material behavior.

[1]- Liu, Z., Wu, C., & Koishi, M. (2019). A deep material network for multiscale topology learning and accelerated nonlinear modeling of heterogeneous materials. Computer Methods in Applied Mechanics and Engineering, 345, 1138–1168.

[2]- Liu, Z., & Wu, C. (2019). Exploring the 3D architectures of deep material network in data-driven multiscale mechanics. Journal of the Mechanics and Physics of Solids, 127, 20–46.

[3]- Gajek, S., Schneider, M., & Böhlke, T. (2020). On the micromechanics of deep material networks. Journal of the Mechanics and Physics of Solids, 142, 103984.

[4]- Matthias Kabel, Dennis Merkert, & Matti Schneider (2015). Use of composite voxels in FFT-based homogenization. Computer Methods in Applied Mechanics and Engineering, 294, 168-188.

[5]- Matthias Kabel, Andreas Fink, & Matti Schneider (2017). The composite voxel technique for inelastic problems. Computer Methods in Applied Mechanics and Engineering, 322, 396-418.

[6]- Schneider, M. (2022). An algorithm for generating microstructures of fiber-reinforced composites with long fibers. International Journal for Numerical Methods in Engineering, 123(24), 6197-6219.

Modeling for the description and prediction of processes in nature often leads to partial differential equations. Solving these field equations can only be done analytically in very few cases, so that in practice numerical approximation methods are often used. Variational methods like the Galerkin method have proven to be very effective and are widely used in industry and research. To set up the system of equations, integration over the area to be calculated is necessary. For more complex geometries or nonlinear equations, analytical integration becomes difficult or even infeasible, so that integration is also often performed numerically in the form of weighted evaluations of the integrand, the Gauss quadrature. In order to benefit from the quasi-optimal accuracy of the Galerkin method according to Cea’s lemma in the linear case, the quadrature scheme must also be of sufficient accuracy. On the contrary, for more complex constitutive laws, under-integration is often used in engineering to save computational time. Based on a split of geometric and material nonlinearities, the present talk introduces a one-point integration scheme that is able to integrate polynomial shape functions of arbitrary order geometrically accurate. The material nonlinearity can be captured with the desired accuracy via a Taylor series expansion from the nonlinear state. As a demonstration the integration scheme is applied to two-dimensional polygonal shaped second order virtual elements where the quadratic projection is integrated via a single integration point.

Metal additive manufacturing (MAM) has received significant attention in recent years due to its significant advantages such as increased design flexibility for complex geometries, shorter production-cycle, and efficient use of raw materials. To fully realize the potential of MAM in the context of Industry 4.0, it is necessary to address challenges related to the mechanical reliability of printed parts and their associated costs. Currently, trial-and-error methods are the most common way of optimizing MAM process conditions for achieving the desired printing quality. Meanwhile, numerical simulations can provide a more profound understanding of the physical phenomena involved in the build process, leading to a more systematic optimization of process conditions, and ultimately making the `first-time-right' high-quality production possible. Achieving a thorough quantitative understanding of the process requires insights from models covering various physical aspects including thermal, mechanical, metallurgical, and fluid-dynamics interactions. However, high-fidelity simulations of such models are accompanied by significant computational costs and therefore have limits in applications, particularly in sensitivity and optimization analyses where solutions for a wide range of scenarios are required.

To address this challenge, we initiated a project in 2021, with the support of the Swiss National Science Foundation (SNSF), to explore the feasibility of meaningful acceleration of these simulations without significant compromise in accuracy and reliability. Specifically, the project aims to develop solutions for thermal, microstructure, and residual stress simulations for the laser powder bed fusion (LPBF) process. To generate experimental validation data, Hastelloy X serve as the 'model material'. An overview of the results obtained so far, focusing on thermal and microstructure simulations, are presented.

Several techniques have been examined to reduce the computational cost of thermal simulation for LPBF, including a multi-scale simulation strategy, surrogate modelling, and physics-informed neural networks (PINNs), where the advantages and limitations of each approach are discussed. In the field of microstructure modelling, a 'Neural Cellular Automata' method has been developed, which outperforms the conventional Cellular Automata with up to 6 orders of magnitude acceleration in computation speed. Moving forward, the project will continue with a focus on the development of affordable high-fidelity models of residual stress development until 2025.

Additive Manufacturing (AM) and especially its metal variants constitute today a reality for the fabrication of high-performance industrial components. In particular, AM allows the construction of novel cellular structures, the so-called lattices, where well-designed unit cells are periodically repeated over a macro-shape to achieve exceptional specific performances, such as unprecedent stiffness-to-weight ratios. These structures, however, are very difficult to simulate numerically: on the one hand, the application of multiscale methods based on homogenization appears delicate due to an insufficient separation of scales (macro versus cell scales); on the other hand, solving directly the high-fidelity, fine-scale problem requires handling large numbers of complex cells which is often intractable if standard methods are blindly used. As a solution, immersed domain techniques have been applied, but such methods, generic in terms of applications, may not be optimal in the case of lattices.

In this context, the purpose of this work is to develop a HPC algorithm dedicated to lattices that takes advantage of the geometric proximity of the different cells in the numerical solution. In order to do so, we start by adopting the CAD paradigm based on spline composition along with its corresponding IGA framework. This offers (i) great flexibility to design any lattice geometry and (ii) fast multiscale assembly of the IGA system. Then, we resort to the family of Domain Decomposition solvers, and develop an inexact FETI based algorithm that avoids solving numerous local cell-wise systems. More precisely, we extract the “principal” local cell stiffnesses using a greedy approach, and use the latter as a reduced basis to efficiently solve all the cell-wise systems. It results in a scalable algorithm that tends to be matrix-free. During the talk, a range of numerical examples in 2D and 3D will be presented to account for the efficiency of our method both in terms of memory and computational cost reduction.

Discontinuous fiber reinforced composites are used in many application areas ranging from automotive to healthcare. Such parts are often manufactured in and injection molding process, as it is an economical process for high volume markets. The simulation of the injection molding process is well established and specific commercial tools have been developed for this task. However, the transient solution of the underlying non-linear multi-phase flow is computationally expensive and computation may take multiple hours for complex geometries. This computational time is prohibitively large for computational optimization of the product design or the process parameters. Hence, we propose a two-step process to accelerate the mold filling prediction: i) Solve a modified Eikonal equation to compute distance maps to the injection gate and nearest walls. This is computationally cheap, as it is only a stationary equation to solve. ii) Train feed forward neural networks to obtain a data-driven relation between the encoded distance maps and mold filling features, such as fill time and fiber orientation. We sample a set of geometries, automatically generate CAD models, and simulate these in a commercial injection molding solver to build a training data set. Subsequently, we apply different feed forward neural network architectures and evaluate their performance.

Phase field modeling has been widely applied to model the evolution of domain patterns in various phase transformation problems. Existing phase-field models for the evolution of domain structures in ferroelectrics are based on an Allen-Cahn-type evolution law. This evolution law successfully captures equilibrium domain structures. However, it fails to capture rate effects due to its assumption of a linear kinetic relation between the thermodynamic driving force acting on a domain wall and the domain wall velocity. To overcome this limitation, we propose a new phase field model for ferroelectrics (Guin and Kochmann, 2022), one that incorporates nonlinearities in the kinetics of domain walls and fully accounts for electro-mechanical coupling. As a multi-phase-field generalization of the model of Alber and Zhu (2013), it is based on the domain volume fraction of each variant as the primary phase field and incorporates the anisotropic dielectric, elastic, and piezoelectric properties of the different variants. This multi-phase field generalization further allows imposing different kinetic relations in different types of domain walls. This new phase field model is validated through a comparison with the target sharp-interface model embedding nonlinear kinetics. With the ability to easily modify these different material properties, we investigate multiphysical effects to the growth of the ferroelectric embryo, and show the open challenge (in common with all ferroelectric phase field models) of the magnitude of the interfacial energy of the regularized domain wall.

Ferroelectrics exhibit many interesting effects, both linear and nonlinear, which is why these materials are widely used in science and industry. Recently, the nonlinear effects have also been employed in the field of energy harvesting [1, 2, 3], while for a long time only linear effects were exploited. Moreover, nonlinear effects are irreversible and are accompanied by energy dissipation, which generally leads to a temperature rise of the material. For modeling the characteristic nonlinear effects of ferroelectric materials, there are various possibilities, in particular microphysical and, phenomenological models.

For describing mutually coupled dissipative processes in ferroelectrics, in particular ferroelectric domain switching and viscoelasticity, a hybrid micromechanical - rheological constitutive model is developed and embedded in the framework of a multiscale modeling approach. The mathematical theory is consistent against the background of rational thermodynamics and deals with two types of internal variables. The advanced modeling approach is applied to identify novel energy harvesting cycles exploiting dissipative effects, resulting in a major electric work output.

REFERENCES

[1] W. Kang, L. Chang and J. E. Huber, Nano Energy 93 (2022), p. 106862.

[2] L. Behlen, A. Warkentin and A. Ricoeur, Smart Mater. Struct. 30 035031 (2021).

[3] A. Warkentin, L. Behlen and A. Ricoeur, Smart Mater. Struct. 10.1088/1361-665X/acafba (2023).

Magneto-active elastomers (MAE) are one of many emerging functional materials. Research applications span mechanical, civil, and biomedical engineering as actuators, sensors, vibration absorbers and vibration isolators. MAE consist of a soft elastomeric matrix filled with small, relatively rigid magnetizable inclusions. Set in a magnetic field, the inclusions deform the microstructure and, at the macro-scale, either stiffen by up to three orders of magnitude or bend to large strains.

Most MAE models focus on magneto-mechanical constitutive relations. This contribution showcases other physical phenomena and their coupled interactions. These phenomena include thermo-mechanical coupling and viscous dissipation leading to heat generation within the material. The model is capable of capturing dynamic effects, particularly when MAE are used as vibration absorbers. Formulated for three-dimensional finite deformations, this model handles incompressible material behavior through a Q1P0 finite element framework.

This study explores the relationship between magnetic fields and deformation in Ferromagnetic Shape Memory Alloys (FSMA), which are materials capable of sensing and actuation. These alloys can exhibit high strains of up to 6% when subjected to a magnetic field. To achieve this goal, a finite deformation formulation approach is proposed based on the multiplicative decomposition of the deformation gradient. In addition, a magneto-thermo-mechanical constitutive model for FSMA is discussed, which is based on a specific Helmholtz free energy function. The evolution equations of the internal magnetic and mechanical state variables are determined using a transformation function, and the model parameters are calibrated under different loading conditions. Finally, the model predictions for FSMA are compared against experimental results.

The renewable energy supply, the independence of fossil resources, as well as the change in mobility act as a driving force on technological innovation. To meet these challenges of our time, new and particularly powerful highperformance magnets are necessary [1], relying on new earth abundant materials and resource efficient processes. It has been shown that composite materials consisting of ferromagnetic grains separated by paramagnetic interphases can contribute to significant improvements in coercivity, when these interphases decouple the magnetic exchange between the individual grains, compare [2]. Novel processing routes based on severe plastic deformations (SPD) or additive manufacturing (AM) can be an option to tailor such magnetic composites. Here, the micromagnetic theory can be applied to numerically predict the magnetization distributions on fine scales. Due to their flexibility, finite elements are well suited to discretize and analyze strongly heterogeneous microstructures [3]. The evolution of the magnetization vectors is described by the Landau-Lifshitz-Gilbert equation, which requires the numerically challenging preservation of the Euclidean norm of the magnetization vectors, see [4,5]. With the aim to correctly reproduce the behavior of magnetic materials, competing energy contributions are considered within the energy functional, which are also responsible for the formation of magnetic domains. Also, grain boundaries, defect layers and misoriented grains can have a huge impact on the macroscopic hysteresis behavior of magnetic materials. Especially magnets formed by SPD are exposed to the potential stress-induced defects that might outweigh their

production benefits. Hence, micromagnetics analyses are performed to estimate the risks and challenges of these novelties.

[1] O. Gutfleisch, et al.: Magnetic Materials and Devices for the 21st Century: Stronger, Lighter, and More Energy Efficient. Advanced Materials, 23, 821–842, (2011).

[2] M. Soderznick, et al.: Magnetization reversal of exchange-coupled and exchange-decoupled Nd-Fe-B magnets observed by magneto-optical Kerr effect microscopy. Acta Materialia, 135, 68–76, (2017).

[3] A. Vansteenkiste, et al.: The design and verification of MuMax3. AIP Advances, 4, 107133, (2014).

[4] A. Prohl: Computational Micromagnetism. Springer, (2001).

[5] M. Reichel, B.-X. Xu and J. Schröder: A comparative study of finite element schemes for micromagnetic mechanically coupled simulation. Journal of Applied Physics, 132, 183903, (2022).

Isogeometric Analysis was proposed more than ten years ago by Tom Hughes et al. to bridge the gap between computer-aided design and the finite element method. The original method uses Non-Uniform Rational B-Splines (NURBS) for the description of geometry and solution in the context of mechanics. Later, Buffa and Vazquez showed how B-Splines can form a De Rham sequence and thus made the methods interesting for multiphysics simulations including electromagnetism. More recently, mortaring and boundary elements methods have been developed, such that there is a large zoo of isogeometric methods available. This presentation will discuss those methods with the aim to optimize electric machines in the context of e-mobility.

In the past two decades, the development of cutting-edge soft-computing technologies and their application to engineering problems has demonstrated huge potential to simulate complex non-linear problems. Machine learning and computer vision can play a significant role in supporting decision-making for infrastructure design, construction, and operation in several ways. They are often use for automated design and real-time optimization, virtual control of the construction process, to support real-time monitoring to make informed decisions regarding asset maintenance, repair, or replacement, estimation of the environmental impacts and optimisation of resources or processes to promote sustainability, and so on.

However, the development of robust prediction tools based on Machine Learning (ML) techniques requires the availability of complete, consistent, accurate, and large datasets. The application of ML in structural engineering has been limited because, although real-size experiments provide complete and accurate data, they are time-consuming and expensive. If we look at large infrastructure projects, the available data is often incomplete and associated with uncertainties or is difficult to interpret. Over the past decades, a vast amount of data has been collected about the condition of our structures and stored in asset management systems in reports, however, this data was collected in an unsystematic manner and often presented in a highly subjective way. The average data scientist spends more than 60% of their time on collecting, organizing, and cleaning data instead of the actual analysis. This is why there is an increasing trend of producing synthetic data. While synthetic data offers benefits compared to real-world data (e.g., increased data quality, scalability and interoperability), it is limited mostly due to bias, lack of realism and accuracy, and the inability to represent the response of complex systems.

In this talk, I will discuss the balance of real-word and synthetic data and how to best leverage the strengths of both to maximise the potential of ML to support decision making for design, construction and maintenance of structures and infrastructure. I will reflect on how ML algorithms and their application in structural engineering have evolved over the past decade, their potential and limitations, and the way forward. Finally, I will present several examples of how ML can be used to optimise structural design [1,2], to virtually control the construction process and minimise the impact on the existing environment [3], and to support visual inspection and maintenance of structures, providing a high level of consistency and automation [4].

References

[1] Cabrera, M., Ninic, J. and Tizani, W., 2023.. Eng with Comp, pp.1-19.

[2] Ninić, J., Gamra, A., Ghiassi, B., 2023. Underground Sp.

[3] Ninić, J., et al., 2017. Tun and Underground SpTech, 63, pp.12-28.

[4] Bush, J. et al., 2021. EG-ICE, Berlin, Germany (pp. 421-431).

Studying the evolution of alloy polycrystalline microstructures under the action of thermomechanical loads such as those occurring during metal additive manufacturing (AM), quenching, welding, laser rescanning, etc., can help identify the impact of different process parameters on the origin of residual stresses, plastic deformations, the eventual mechanical response. This information can be used to guide the aforementioned processes to design microstructures with a desired response.

To that end, a polycrystal thermo-elasto-viscoplastic finite element (T-EVP-FE) model has been developed. It takes into account the strong coupling between evolving temperatures and stresses under thermomechanical boundary conditions. The constitutive laws of the model include the generalised 3D Hooke’s law, a viscoplastic power law that accounts for the shear rate from each slip system, a Voce-type hardening law, and the generalised Fourier law of heat conduction.

Recently, a series of laser scanning experiments have been performed using a novel laser-SEM (scanning electron microscope) experimental setup; this device is a coupling between a continuous wave fibre laser and an environmental SEM. In these experiments, electron backscattered diffraction was performed before and after laser scanning to study the role of laser scanning on an AM 316L stainless steel microstructure. The experiment revealed the formation of misorientation bands, and hence, geometrically necessary dislocations, that vary as a function of the laser scanning velocity. The T-EVP-FE model has been applied to simulate these laser scanning experiments.

In this talk, the model will be presented, the microstructure state will be compared with experimental observations and the role of laser scan velocity on the evolution of intergranular residual stresses, plastic deformation, stress concentrations, geometrically necessary dislocation formation, etc. will be discussed.

The solution of multi-field problems and the numerical implementation by means of the finite element method constitute a sophisticated part of the characterisation of complex material behaviour. Particularly the implementation into commercial finite element codes is of major importance for practical and industrial applications. Although the wide range of available finite element codes (e.g. Abaqus) provides the opportunity for multiphysical modelling, those implementations are rather restricted to the solution of two coupled field equations. In [1, 2] an Abaqus UMAT framework was introduced to use the balance of linear momentum and the heat equation for the solution of two arbitrary coupled field equations of Laplace-type. An extension of the framework to the solution of three coupled Laplace equations is presented in this contribution.

A comprehensive implementation framework for such a three-field problem into the finite element software Abaqus is provided. The procedure is derived for a micromorphic approach in thermomechanics. Although the provided framework contributes to a particular three-field problem, it is not limited to a particular application or a specific number of coupled field equations from a conceptual point of view. The solution of the considered system of equations is separated onto two coupled domains and is based on a two-instance formulation.

To assess the framework for a particular constitutive model, a gradient-enhanced damage model in a thermo-mechanical setting is adopted and representative simulation results are discussed on a local and a global level. Since the framework is not limited to the solution of three coupled field equations, the extension to arbitrary multi-field problems is discussed.

[1] Ostwald R., Kuhl E., Menzel A. (2019) On the implementation of finite deformation gradient-enhanced damage models. Computational Mechanics 64(847-877). https://doi.org/10.1007/s00466-019-01684-5.

[2] Seupel A., Hütter G., Kuna M. (2018) An efficient FE-implementation of implicit gradient-enhanced damage models to simulate ductile failure. Engineering Fracture Mechanics 199:41-60. https://doi.org/10.1016/j.engfracmech.2018.01.022.

High concentrations of vacancies in crystals may be the result of large plastic deformations or irradiation. Void formation and subsequent growth are well-known to be involved in swelling of irradiated materials and seem to play an important role for the nucleation and evolution of porosity in the early stages of ductile failure as recent experiments suggest [1]. Vacancy diffusion and void formation have been modelled using spatially resolved approaches like the phase-field method. Taking into account that vacancies induce an eigenstrain field, which emerges from the relaxation of the surrounding crystal lattice if a single atom is removed, indicates that the evolution of vacancy concentration needs to be properly coupled to the elastic stress field.

In our recent work [2], we proposed a model for coupling elastically driven vacancy diffusion with a phase-field model of void surfaces, which overcomes the short-comings of former models and closely reproduces the sharp interface solution for small-strain elasticity. This is achieved by making the vacancy eigenstrain a function of the non-conserved order parameter used to distinguish the void and crystal phase. With the recent findings and aiming at being able to numerically analyze the early stages of ductile failure as implied by the mentioned experiments, we present the extension of our model to finite strains. Using a multiplicative split for the deformation gradient, a proper coupling of kinematics and the kinetics of vacancy–void interactions is emphasized. A thermodynamically consistent definition of the energy contributions and the derivation of the resulting driving forces based on the underlying phase-field description are outlined. The model is verified with benchmark problems and the influence of the chemo-mechanical coupling is discussed. The implementation of the governing equations in the multi-physics software tool DAMASK [3] allows a coupling to different plasticity laws, like e.g. continuum dislocation dynamics theory for modelling creep or ductile failure.

[1] P. J. Noell et al. Nanoscale conditions for ductile void nucleation in copper: Vacancy condensation and the growth-limited microstructural state. Acta Materialia, 184:211–224

[2] K. A. Pendl and T. Hochrainer. Coupling stress fields and vacancy diffusion in phase-field models of voids as vacancy phase. [Accepted for publication in Computational Materials Science]

[3] F. Roters et al. DAMASK – The Düsseldorf Advanced Material Simulation Kit for modeling multi-physics crystal plasticity, thermal, and damage phenomena from the single crystal up to the component scale. Computational Materials Science, 158:420–478

In the present work, a variational formulation for coupled chemo-mechanical problems in elastic and dissipative solids at infinitesimal strains is outlined. In doing so, it is seen that the gradient of the primary fields additionally enter the energetic and dissipative potential functions, resulting in additional balance equations. The governing balance equations of the coupled problem are derived as Euler equations of the incremental variational principles, formulated in a continuous-and discrete-time setting. Furthermore, the variables governing the inelastic process are locally condensed which yields a reduced global problem that is solved in a discrete-space-time setting. The symmetric structure of the proposed framework with respect to the primary and state variables is an advantage, and this is exploited in the numerical treatment within the finite element paradigm. The framework is applied to Cahn-Hilliard- type diffusion and Allen-Cahn-type phase transformation in elastic and dissipative solids. The applicability of the proposed framework is demonstrated by means of two- and three-dimensional representative numerical simulations.

In this contribution, we give an overview of methods and techniques

that we apply in our group to elucidate the specific behaviour

of functional nano-structures in the condensed phase on a molecular basis.

A special focus will be on materials that can be reversibly photoswitched by external light stimulus.

When molecular photo-switches, such as azobenzene or norbornadiene, are embedded into a sufficiently soft polymer matrix the resulting compound can undergo a mechanical deformation induced by light of a specific wavelength. These photo-sensitive compounds have the potential to be applied as soft actuators without the need for hard wired electronics or a separate energy source. Such characteristics are especially attractive in the design of micro-scale robots but also other applications such as high-speed data transfer or the conversion of photonic energy into a mechanical response holds great promise.

Despite these almost futuristic possibilities, photo-sensitive polymers have not yet experienced a sufficient attention in industrial applications. One important factor to increase the acceptance of this group of soft smart materials is the formulation of a rigorous constitutive modeling approach in combination with numerical simulation methods. Thus, in this contribution we present a photo-mechanical modeling approach solved with the help of a finite element implementation.

We develop a theoretical and computational framework to perform topology optimization of the representative volume element (RVE) of flexoelectric metamaterials [2].

The flexoelectric effect is an electromechanical coupling between polarization and strain gradient as well as strain and electric field gradients, present in small (micro-to-nano) scales [1]. It is universal to dielectrics, but, as compared to piezoelectricity, it is more difficult to harness as it requires small scales and field gradients. These drawbacks can be overcome by suitably designing geometrically polarized metamaterials made of a nonpiezoelectric base material but exhibiting apparent piezoelectricity [3].

We solve the governing equations of flexoelectricity on a high-order generalized-periodic Cartesian B-spline approximation space. The geometry is unfitted to the mesh, and described by a periodic level set function. Genetic algorithms are considered for the multi-objective optimization of the RVE topology, where area fraction competes with four fundamental piezoelectric functionalities (stress/strain sensor/actuator). During the optimization process, the RVE topologies are restricted to be fully-connected in a single group of material.

We obtain Pareto fronts and discuss the different material topologies depending on the area fraction and the apparent piezoelectric coefficient being optimized. Overall, we find RVE topologies exhibiting a competitive apparent piezoelectric behavior as compared to reference piezoelectric materials such as quartz and PZT ceramics. This opens the possibility to design a new generation of devices for sensing, actuation and energy harvesting application using a broad class of base materials.

References

[1] D. Codony, A. Mocci, J. Barceló-Mercader, and I. Arias: Mathematical and computational modeling of flexoelectricity. Journal of Applied Physics 130(23) (2021), 231102.

[2] F. Greco, D. Codony, H. Mohammadi, S. Fernández-Méndez, and I. Arias: Topology optimization of flexoelectric metamaterials with apparent piezoelectricity. arXiv preprint arXiv:2303.09448 (2023).

[3] A. Mocci, J. Barceló-Mercader, D. Codony, and I. Arias: Geometrically polarized architected dielectrics with apparent piezoelectricity. Journal of the Mechanics and Physics of Solids 157 (2021), 104643.

One phenomenon to consider in metal forming nowadays is the accumulation of ductile damage during the forming process. Ductile damage, i.e. the nucleation, growth and subsequent accumulation of micro-defects such as voids, is inherently present in any formed part. Therefore, it is advisable to reduce these damage effects and in turn produce parts with reduced damage accumulation and thus higher safety factors. Herein, optimisation is a very useful tool to enhance already established processes with damage minimisation in mind. By defining process dependent parameters as the design variables of the optimisation, improved tool sets can be generated to reduce the ductile damage of the formed part.

An important aspect to consider when dealing with simulation of forming processes is the underlying necessity of contact algorithms. Approaches to handle these discontinuous problems are available in literature, by e.g. utilising sub-gradients, however, their application mainly see academic use. The problems for the proposed optimisation are however very complex in nature and therefore require robust and efficient implementation. Consequently, the commercial finite element software Abaqus is used to simulate the processes and solve the necessary contact problems.

In this submission, a framework defined in Matlab is presented, which utilises Abaqus as the finite element program to solve the stated optimisation problems. The framework is applied to different forming processes, such as rod extrusion and deep drawing-like processes, in order to optimise them with regard to their accumulated damage. Different sets of geometric parameters are defined, which in turn result in optimal design for the work tools of the analysed problems. Due to the nature of the framework, the optimisation is not limited to process optimisation and further examples regarding curve fitting for experimental setups are also presented.

When molding processes, such as injection molding, are used to produce plastics parts, it can be difficult to achieve the correct product shape. As part of the process, the material must cool down and solidify. Since this can happen in an inhomogeneous way, residual stresses can remain in the material. These lead to warpage, after the part is ejected from the machine.

There are several aspects of the process that could be adjusted to improve the resulting product shape. The focus of this work is on the shape of the mold cavity. If suitable adjustments are made to this cavity, the product shape can be improved although shrinkage and warpage still occur. In order to estimate the effects of certain cavity shape changes, a numerical simulation method for the process is required.

This cavity design problem can then be treated either as a shape optimization problem or as an inverse problem. In the former case, a suitable shape parameterization and objective function need to be found. Both options profit from the use of splines, since this allows the shape to be transferred back to a CAD format. The method of Isogeometric Analysis (IGA) offers a convenient way of using splines as a geometry representation in the Finite Element Method. We will discuss the different design approaches and explain the benefits and challenges involved with the spline representations.

In the typical product development process of an automotive part, multiple disciplinary teams collaborate to converge on a final design. Structural mechanics, design, crashworthiness and manufacturability are relevant disciplines that mutually influence one another. Sheet metal forming operations are the cornerstone of automotive part production, as a significant portion of the individual components of the Body-in-White (BiW) are fabricated through stamping and deep-drawing processes. Manufacturability assurance for sheet metal forming is commonly addressed by engineering experience and heuristic rules based on geometrical constraints. This work explores the idea of formulating analytical manufacturing constraints for stamped and deep-drawn parts and its inclusion into existing multidisciplinary shape optimization workflows to address formability and performance objectives simultaneously.

As discussed by [1], gradient methods based on adjoint sensitivity analysis, together with a filtering technique as Vertex-Morphing are powerful tools for the typical large and very large optimization use cases in the industry. In this contribution, we present the current progress in the formulation of a constraint for shape optimization that accounts for the manufacturing process, discuss the definition of a meaningful objective function and present details regarding the calculation of adjoint-based sensitivities and its combination with Vertex-Morphing. The formulations of the primal and adjoint problems are also presented, based on the simplified Finite Element Analysis for sheet metal forming proposed by [2].

[1] Kai-Uwe Bletzinger. A consistent frame for sensitivity filtering and the vertex assigned morphing of optimal shape. Structural and Multidisciplinary Optimization, 49, 01 2014.

[2] Y. Q. Guo, J. L. Batoz, J. M. Detraux, and P. Duroux. Finite element procedures for strain estimations of sheet metal forming parts. International Journal for Numerical Methods in Engineering, 30(8):1385–1401, 1990.

This work aims to gain a deeper understanding of sensitivity information through the use of principal component analysis (PCA). By decomposing sensitivity matrices, it is possible to explore and analyze the underlying relationships between variables and the impact of their changes on the overall structure. The approach for this analysis is discussed in [1]. PCA allows us to analyze the eigenvectors of the covariance matrix, which are known as the principal components. The first principal component is considered the most significant mode of variation as it indicates the direction with the highest variance in the data. Similarly, the second principal component represents the direction with maximum variance, but this time it must be orthogonal to the first principal component. This process continues for the remaining principal components. The work at hand makes use of gradient-based sensitivity analysis [2] for dynamic structures and compares two different methods for shape design: Isogeometric Analysis (IGA) [3] and Finite Element Method (FEM). The main focus is on using direct differentiation, but if analytical gradients are not available, numerical differentiation methods such as complex-step method (CSM) can be used as alternatives. We utilize different types of basis functions, such as Bernstein polynomials, B-Splines, and Non-Uniform Rational B-Splines (NURBS), to describe the shape of the structure. IGA has several advantages over traditional FEM-based approaches. These advantages include the ability to accurately describe geometry using fewer control points, high-order continuity, and increased flexibility due to control point weights. These characteristics have a significant impact on shape sensitivity analysis. IGA is used during the structural optimization process to avoid costly remeshing and design velocity field calculations. It is more efficient and effective than traditional FEM approaches for these tasks. In contrast to static analysis, the response of a structure to time-dependent loads is significantly affected by inertia and damping effects. The necessary computational characteristics for this type of problem are discussed and the full solution algorithm is presented.

References

[1] N. Gerzen and F.-J. Barthold, “Design space exploration based on variational sensitivity analysis,” PAMM, vol. 14, no. 1, pp. 783–784, Dec. 2014. DOI: 10.1002/pamm.201410374.

[2] F.-J. Barthold, N. Gerzen, W. Kijanski, and D. Materna, “Efficient variational design sensitivity analysis,” in Mathematical Modeling and Optimization of Complex Structures (Computational Methods in Applied Sciences), P. Neittaanmäki, S. Repin, and T. Tuovinen, Eds., Computational Methods in Applied Sciences. DOI: 10.1007/978-3-319-23564-6_14.

[3] T. Hughes, J. Cottrell, and Y. Bazilevs, “Isogeometric analysis: Cad, finite elements, nurbs, exact geometry and mesh refinement,” Computer Methods in Applied Mechanics and Engineering, vol. 194, no. 39-41, pp. 4135–4195, 2005. DOI: 10.1016/j.cma.2004.10.008.

Topology optimization is an effective numerical tool to design high-performance, efficient and economical lightweight structures. In this talk the solution procedure for a two material topology optimization problem constrained by a scalar second order PDE is presented. The approach relies on a numerical topological-shape derivative as a main ingredient for the gradient-based solution algorithm.

We state the optimization problem in the continuous setting and subsequently discretize it. On the continuous level we review the classical shape derivative where the perturbation is realized by the action of a vector field and the classical topological derivative where the perturbation is done by means of sets. In contrast to this, in the presented approach the geometry is represented by the zero level-set of a scalar function. Based on this representation we suggest a topological-shape derivative unifying the concepts of shape derivative and topological derivative. In a next step we consider the discretization of the PDE as well as the level-set function by linear triangular Lagrange finite elements. In this numerical setting we can now consider the perturbation of the level-set function by the perturbation of its nodal values. Based on this we give explicit formulas for the computation of the numerical topological-shape derivative. This derivative information is used in an algorithm to update the level-set function where no distinction between shape changes and topological changes is made. The algorithm is tested in a numerical example, where the shape of two circles with different radii is recovered.

The growing field of in silico medicine is focusing mostly on the two largest classes of medicinal products: medical devices and pharmaceuticals. However, also for advanced therapeutic medicinal products, which essentially combine medical devices with a viable cell or tissue part, the in silico approach has considerable benefits. In this talk an overview will be provided of the budding field of in silico regenerative medicine in general and computational bone tissue engineering (TE) in particular. As basic science advances, one of the major challenges in TE is the translation of the increasing biological knowledge on complex cell and tissue behavior into a predictive and robust engineering process. Mastering this complexity is an essential step towards clinical applications of TE. Computational modeling allows to study the biological complexity in a more integrative and quantitative way. Specifically, computational tools can help in quantifying and optimizing the TE product and process but also in assessing the influence of the in vivo environment on the behavior of the TE product after implantation. Examples will be shown to demonstrate how computational modeling can contribute in all aspects of the TE product development cycle: from providing biological blueprints, over guiding cell culture and scaffold design, to understanding the etiology and optimal treatment strategies for large skeletal defects. Depending on the specific question that needs to be answered the optimal model systems can vary from single scale to multiscale. Furthermore, depending on the available information, model systems can be purely data-driven or more hypothesis-driven in nature. The talk aims to make the case for in silico models receiving proper recognition, besides the in vitro and in vivo work in the TE field.

The interaction between slender fiber- or rod-like components, where one spatial dimension is much larger than the other two, with three-dimensional structures (solids) is an essential mechanism of mechanical systems in numerous fields of science, engineering and bio-mechanics. Examples include reinforced concrete, supported concrete slabs, fiber-reinforced composite materials and the impact of a tennis ball on the string bed of a tennis racket. Applications can also be found in medicine, where stent grafts are a commonly used device for endovascular aneurysm repair, and in many biological systems such as arterial wall tissue with collagen fibers.
The different types of dimensionality of the interacting bodies, i.e., slender, almost one-dimensional fibers and general three-dimensional solids, pose a significant challenge for typical numerical simulation methods. The presented focuses on developing novel computational approaches to simulate the interaction between these fiber-like structures and three-dimensional solids. The key idea is to model the slender components as one-dimensional Cosserat continua based on the geometrically exact beam theory, enabling an accurate and efficient description of the fibers. This results in a mixed-dimensional beam-to-solid interaction problem.
In a first step positional and rotational coupling between the beam centerline and the underlying solid in line-to-volume problems are addressed. Mortar-type methods, inspired by classical mortar methods from domain decomposition or surface-to-surface interface problems, are used to discretize the coupling constraints. A subsequent penalty regularization eliminates the Lagrange multipliers from the global system of equations, resulting in a robust coupling scheme that avoids locking effects. Furthermore, consistent spatial convergence behavior, well within the envisioned application range, is demonstrated.
In a second step, the previously developed algorithms for line-to-volume coupling are extended to to line-to-surface coupling. This introduces the additional complexity of having to account for the surface normal vector in the coupling constraints. Consistent handling of the surface normal vector leads to physically accurate results and guarantees fundamental mechanical properties such as conservation of angular momentum.
Finally, a Gauss point-to-segment beam-to-solid surface contact scheme that allows for the modeling of unilateral contact between one-dimensional beams and two-dimensional solid surfaces is presented.
The previously mentioned building blocks constitute a novel mixed-dimensional beam-to-solid interaction framework, which is verified by theoretical discussions and numerical examples. Already in the present state, the presented framework is an efficient, robust, and accurate tool for beam-to-solid interaction problems and can become a valuable tool in science and engineering.

We recently proposed an approach for Efficient Unsupervised Constitutive Law Identification and Discovery (EUCLID), which exploits machine learning tools such as sparse regression [1–3], Bayesian learning [4], or neural networks [5] to automatically discover material laws independent of stress data, but solely based on full-field displacement and global force data obtained from mechanical testing. The displacement field can be measured on the surface of a target specimen via digital image correlation (DIC).

An important feature of the approach is that, in principle, the discovery of the material law can be performed in a one-shot fashion, i.e., using only one experiment. However, this capability heavily relies upon the richness of the measured displacement data, i.e., their ability to probe the stress-strain space (where the stresses depend on the constitutive law being sought) to an extent sufficient for an accurate and robust discovery process. The richness of the displacement data is in turn directly governed by the specimen geometry.

In the present study, we aim to optimally design the geometry of the target specimen in order to maximise the richness of the deformation field obtained by the DIC method. We seek to excite various deformation modes (from tension/compression to shear) in a single optimised specimen, to maximise the performance of EUCLID. To this aim, we utilise density-based topology optimisation driven by an objective function deduced from EUCLID itself, which essentially aims at enhancing the identification robustness of material parameters (especially in noisy measurements). In this contribution, we shed light on the objective function, the topology optimisation framework, and the initial results.

The intricate and nonlinear nature of material behavior, characterized by a progressively changing stress-strain relationship, is a fundamental and indispensable property governing the behavior of numerous mechanical systems. Examples of these mechanical systems include rubber components in automobile suspensions and engine mounts, soft tissues and organs in bio-mechanics and medical engineering, as well as packaging materials such as foam, paper, and plastics. Components used for the construction of these mechanical systems often need to meet specific stiffness requirements which can be influenced by the composition of the employed material, see Steinbrecher et al. [Computational Mechanics, 69 (2022)].

On the macro or micro level, such materials can often be classified as fiber-reinforced materials, i.e., thin and long fibers embedded inside a matrix material. One way to control the stress-strain relationship of fiber-reinforced materials is to alter the geometry of the reinforcements, thus creating passive materials with a highly nonlinear stress-strain response. This can be a viable method for the development of optimized system components or meta-materials.

This method can be explored with a single beam embedded in a softer matrix. If the embedded beam is straight, the stress increase would be approximately linear with increasing strain. Bending the beam inside the matrix will lower the starting stress rate. The stress rate increases until the encased beam is straight, at which point the stress rate will not increase further. By manipulating the initial geometry of the beam, the evolution of the strain rate can be influenced.

Previously, Reinforcement Learning based shape optimization has been used to optimize structures in the context of fluid dynamics, see Fricke et al. [Advances in Computational Science and Engineering, 1 (2023)]. This approach is different from classical optimization methods, as it trains an agent to solve a specific task inside a defined problem set. While the training is computationally more expensive than a single optimization, the trained agent is able to optimize a problem inside the learned problem set with less effort.

Applying the RL-based shape optimization method to the beam geometry, an agent is trained to identify optimal beam geometries for a set of starting stress rates and ending stress rates.

In injection molding, shrinkage and warpage lead to shape deviations of the produced parts with respect to the cavity. Caused by shrinkage and warpage, these deviations occur due to uneven cooling and internal stresses inside the part. One method to mitigate this effect is to adapt the cavity shape to the expected deformation. This deformation can be determined using appropriate simulation models, which then also serve as a basis for determining the optimal cavity shape.

Shape optimization usually requires a sequence of forward simulations, which can be computationally expensive. To reduce this computational cost, we use Gaussian Process Regression (GPR) as a surrogate model. The GPR learns the objective function, which measures the shape difference. This difference is computed by taking the average Euclidean distance of sample points on the surface of the deformed product on the one hand and the desired shape on the other hand. Additionally, GPR has the benefit that it allows to account for uncertainty in the model parameters and thus provides a means to investigate their influence on the optimization result. We present a GPR trained with samples from a finite-element solid-body model. It predicts the deformation of the product after solidification and, together with GPR, allows for efficient cavity shape optimization. The optimization parameters are the position of spline points representing the geometry.

To further improve the learning efficiency, we use Bayesian optimization. This approach selects the next training points by utilizing an acquisition function that balances exploration and exploitation. Here, training points with low function values and high uncertainty are given priority. This achieves the goal of finding the optimal solution with the least number of training points. The Bayesian optimization framework was implemented in Python code.

The material parameters can underlie fluctuations because of different batches or when using recycled material. To account for these uncertainties in the material parameters, a new formulation of the objective function is proposed to find the optimal shape resulting in low shrinkage and warpage as well as low variance with respect to the input parameters.

The nasal cavity is one of the most important organs of the human body. Its various functionalities are essential for the well-being of the individual person. It is responsible for the sense of smell, supports degustation, and filters, tempers, and moistens the inhaled air to provide optimal conditions for the lung. Diseases of the nasal cavity like chronic rhinosinusitis, septal deviation, or nasal polyps may lead to restrictions or complete loss of these functionalities. A decreased respiratory capability, the development of irritations and inflammations, and lung diseases can be the consequences.

The shape of the nasal cavity varies from person to person with stronger changes being present in pathological cases. A decent analysis on a per-patient basis is hence crucial to plan for a surgery with a successful outcome. Nowadays diagnostic methods rely on morphological analyses of the shape of the nasal cavity. They employ methods of medical imaging such as computed tomography (CT) or magnetic resonance imaging (MRI), and nasal endoscopy. Such methods, however, do not cover the fluid mechanics of respiration, which are essential to understand the impact of a pathology on the quality of respiration, and to plan for a surgery. Only a meaningful and physics-based diagnosis can help to adequately understand the functional efficiency of the nasal cavity, to quantify the impact of different pathologies on respiration, and to support surgeons in decision making.

Physics-based methods to diagnose pathologies in the human respiratory system have recently evolved to include results of computational fluid dynamics (CFD) simulations. In the current study, a reinforcement learning (RL) algorithm that learns to optimize the nasal cavity shape based on feedback from CFD simulations is developed. The final structure of the airway then functions as the proposed surgery. It is investigated how the algorithm finds the optimal structural modification based on various optimization criteria:

(i) the capability of inhaling and supplying the lung with air, expressed by the pressure difference between the nostrils and the pharynx;

(ii) the functionality of the nose for heating up incoming air, represented by the temperature difference between the nostrils and the pharynx;

(iii) the possibility for a balanced air supply between the left and right nasal passages, realized by equal mass flow rates through both nasal passages;

Furthermore, different types of RL algorithms are employed and their computational efficiency is analyzed. It is the aim to further advance these techniques to include them into clinical pathways, thereby allowing personalized analyses on a per-patient basis.

With an aim to fill the gaps in the current 3D‐printing technology to digitally fabricate medical assistive devices with significant user benefit, well‐being, and availability; we develop a design methodology that enables the optimization of lightweight multi-material lattice structures in order to enhance the design of prostheses and rehabilitation devices. This is done by firstly developing a suitable multi-variable mathematical model for topology optimization of two-scale structures and secondly demonstrating it on an outer shaft of prostheses (lower limb prostheses shaft). We develop a two-scale gradient-based optimization algorithm procedure of multiple design variables that generates functionally graded structures having excellent performance. Our design methodology employs three families of predefined micro-structures that share similar geometric features. Those two additional families thwart the convergence of our gradient-based algorithm to the global minima and we aim at presenting a computational framework that enhances multi-variable optimizations by avoiding the unfavorable local minima.

Recently, there has been a development of innovative materials that imitate the strong and lightweight properties of natural structures, such as bones, honeycombs and sponges. These materials have a porous microstructure that alternates between solid and void, and are being used in various fields, especially in healthcare, thanks to advanced manufacturing techniques like 3D printing. However, the production time of 3D printed objects can vary depending on factors such as material rigidity, infill pattern, and printing parameters. To address this issue, a computational tool was developed, integrating numerical homogenization and topological optimization in ANSYS Mechanical. The study used computational homogenization to simulate the mechanical properties of the insoles' infill, investigating various infill patterns in terms of mechanical properties and printing performance. The calculated properties were assigned to the insoles' geometries, and different loading scenarios were analysed, considering therapeutic and usage frameworks. Using the results of these structural simulations, several topology optimization analyses were performed with the objective of reducing the frontal part of the insole's compliance while staying within a specified mass threshold. The study aimed to find a distribution of mass that minimized material use and printing time while maintaining a satisfactory structural response during insole insertion into the shoe. Additionally, this computational approach can optimize the material distribution in various orthopaedic devices, making 3D printing production more effective and reducing printing time.

Through recent advances in modern production techniques, particularly in the field of additive manufacturing, new previously unthinkable geometries have become feasible. This vast realm of new possibilities cannot be adequately addressed by classical methods in engineering, which is why numerical design techniques are becoming more and more valuable. In this context, this work aims to present concepts that exploit the emerging possibilities and facilitate numerical optimization.

The numerical optimization is built on a microstructured grid, where the geometry is constructed by means of functional composition between splines, resulting in a regular pattern of building blocks. Here, a macro-spline defines the outer contour, a micro-geometry sets the individual tiles and a parameter-spline controls the local parametrization of the microstructure, e.g., acting on the thickness or material density in a specific region. This approach opens up a broad design space, where the adaptivity of the resulting microstructure can be easily extended by increasing the number of control variables in the parameters-spline via h- or p-refinement. The geometric representation uses volume splines, on the one hand providing full compatibility with CAD/CAM and on the other hand facilitating the use of Isogeometric Analysis (IGA). To fully utilize the potential of this type of geometry parameterization, gradient-based optimization algorithms are employed in combination with analytical derivatives of the geometry and adjoint methods.

We will present first results in two fields of application, namely passive heat regulation and an elasticity problem. Here, we demonstrate how optimized microstructures can compensate for irregular boundary conditions and how compliance can be minimized using these lattice-like structures for major weight reduction.

This research has been supported by European Union's Horizon 2020 research and innovation program under agreement No. 862025.