Inverse design problems are common in engineering and materials science. The forward direction, i.e., computing output quantities from design parameters, typically requires running a numerical simulation, such as a FEM, as an intermediate step, which is an optimization problem by itself. In many scenarios, several design parameters can lead to the same or similar output values. For such cases, multi-modal probabilistic approaches are advantageous to obtain diverse solutions. A major difficulty in inverse design stems from the structure of the design space, since discrete parameters or further constraints disallow the direct use of gradient-based optimization. To tackle this problem, we propose a novel inverse design method based on diffusion models. Our approach relaxes the original design space into a continuous grid representation, where gradients can be computed by implicit differentiation in the forward simulation. A diffusion model is trained on this relaxed parameter space in order to serve as a prior for plausible relaxed designs. Parameters are sampled by guided diffusion using gradients that are propagated from an objective function specified at inference time through the differentiable simulation. A design sample is obtained by backprojection into the original parameter space. We develop our approach for a composite material design problem where the forward process is modeled as a linear FEM problem. We evaluate the performance of our approach in finding designs that match a specified bulk modulus. We demonstrate that our method can propose diverse designs within 1

Obtaining high-quality particle distributions for stable and accurate particle-based simulations poses significant challenges, especially for complex geometries. We introduce a preprocessing technique for 2D and 3D geometries, optimized for smoothed particle hydrodynamics (SPH) and other particle-based methods. Our pipeline begins with the generation of a resolution-adaptive point cloud near the geometry's surface employing a face-based neighborhood search. This point cloud forms the basis for a signed distance field, enabling efficient, localized computations near surface regions. To create an initial particle configuration, we apply a hierarchical winding number method for fast and accurate inside-outside segmentation. Particle positions are then relaxed using an SPH-inspired scheme, which also serves to pack boundary particles. This ensures full kernel support and promotes isotropic distributions while preserving the geometry interface. By leveraging the meshless nature of particle-based methods, our approach does not require connectivity information and is thus straightforward to integrate into existing particle-based frameworks. It is robust to imperfect input geometries and memory-efficient without compromising performance. Moreover, our experiments demonstrate that with increasingly higher resolution, the resulting particle distribution converges to the exact geometry.

Powder-scale simulation of laser powder bed fusion (LPBF) is increasingly vital for understanding, predicting, and controlling metallurgical defects. However, the complex multi-physics and multimaterial interactions involved, along with high computational demands, pose significant challenges. This study presents the first multiphase smoothed particle hydrodynamics (SPH) simulation framework for LPBF under both low and high evaporation regimes, incorporating a multiresolution particle strategy and ray tracing (RT). An adaptive particle refinement (APR) method, compatible with multiphase SPH and optimized for GPU acceleration, is developed to enhance computational efficiency of the multiphase model. The RT model is also optimized and integrated with the APR-GPU architecture, further improving performance. A physics-based wetting force model is introduced, along with a novel method for improving normal vector accuracy near the contact line. The proposed framework is validated through three benchmark cases and applied to simulate LPBF processes. The results demonstrate that the framework achieves high accuracy and efficiency in resolving key LPBF phenomena, including melt pool dynamics and keyhole formation.

In the automotive industry, aluminum alloys are increasingly favored over steel due to their superior strength-to-weight ratio, which benefits fuel efficiency and vehicle performance. However, research on aluminum resistance spot welding is less developed than on steel, leaving knowledge gaps in understanding the process. This study investigates the dynamic electrical resistance during resistance spot welding for two aluminum alloys, EN AW-5182 and EN AW-6014, addressing two gaps through machine learning, big data analysis, and leveraging knowledge drawn from steel resistance spot welding. The study focuses on two aspects. First, we use symbolic regression to identify a mathematical function describing the initial decay of dynamic resistance during the preheating. The resulting formula captures the behavior of the electrical resistance, describing an initial rapid increase proportional to time squared t^2 followed by an exponential decay following e^-t . Second, we develop a machine learning model to predict a critical point in the dynamic resistance at the end of the preheating. The model is trained with an aggregated data set of over 90,000 weld spots. The influence of the features is interpreted with Shapley additive explanations. It is found that material thickness and electrode force are the main factors affecting resistance at the end of preheating. Over a two-sheet material thickness range of 2 to 6 mms, the resistance increases by approximately 8 μΩ , while over an electrode force range of 5 to 8 kN, the resistance decreases by 3.5 μΩ .

High-voltage cables, as applied in the electro-mobility, are highly complex structures regarding their vibration behaviour. The high complexity leads to considerable uncertainty in models for a finite element method (FEM) simulation, which is shown, for example, in the contact modelling between the strands of the cable. To handle this uncertainty and model the structural dynamic, a nonparametric probabilistic approach (NPPA) with random matrices is used for the first time on high-voltage cables. This novel application of NPPA has an advantage over typical FEM analysis by using a more manageable simulation model and eliminating the need for a complex deterministic simulation model. Initially, the NPPA is analysed and enhanced, with an optimization for the dispersion parameter and a frequency shift introduced as methodological improvements. These enhancements result in a comparable scatter band of the frequency response. Following preliminary studies, the cable's dynamic behaviour is examined through experimental modal analysis, after which the dispersion parameters are computed. The NPPA is then applied to the simplified deterministic model with the calculated dispersion parameters, and a Monte Carlo simulation is done. As a result of this simulation, a scatter band is given. The results from the simulation are then compared to the results of an experiment. It is shown that the frequency response from the experiment is almost always in the inner area of the scatter band. Consequently, this innovative method can be used for a risk evaluation according to the path of the frequency response function and an evaluation of the structural behaviour.

In this work, we propose an efficient smoothed particle hydrodynamics (SPH) method for simulating laser powder bed fusion (LPBF). The multi-physics process of LPBF, including the heat transfer and phase change with complex boundaries, is accurately resolved by a novel heat source model and a modified continuous surface force based on a corrected surface delta function. Moreover, we also develop an efficient tensile instability control algorithm for preventing the pressure oscillations. The present method is implemented in a GPU-accelerated framework, and its performance is well demonstrated by simulating the LPBF processes with both single-layer and multi-layer powder beds (with the help of surface reconstruction). The numerical results are compared well with the experimental ones which clearly verify the ability of the present method in capturing the complex physical phenomenon of LPBF.

This book provides a detailed instruction to virtually reproduce the processes of Additive Manufacturing on a computer.

This work presents a meshfree particle scheme designed for arbitrary deformations that possess the accuracy and properties of the Finite-Element-Method. The accuracy is maintained even with arbitrary particle distributions. Mesh-based methods mostly fail if requirements on the location of evaluation points are not satisfied. Hence, with this new scheme not only the range of loadings can be increased but also the pre-processing step can be facilitated compared to the FEM. The key to this new meshfree method lies in the fulfillment of essential requirements for spatial discretization schemes. The new approach is based on the correspondence theory of Peridynamics. Some modifications of this framework allows for a consistent and stable formulation. By applying the peridynamic differentiation concept, it is also shown that the equations of the correspondence theory can be derived from the weak form. Likewise, it is demonstrated that special moving least square shape functions possess the Kronecker- δ property. Thus, Dirichlet boundary conditions can be directly applied. The positive performance of this new meshfree method, especially in comparison to the Finite-Element-Method, is shown in the calculation of several test cases. In order to guarantee a fair comparison enhanced finite element formulations are also used. The test cases include the patch test, an eigenmode analysis as well as the investigation of loadings in the context of large deformations.

This paper presents a parallel implementation for the Optimal Transportation Meshfree (OTM) method on large CPU clusters. Communications are handled with the Message Passing Interface (MPI). The Recursive Coordinate Bisection (RCB) algorithm is utilized for domain decomposition and for implementing dynamic load-balancing strategy. This work involves three new concepts to reduce the computational efforts: Dynamic halo regions, Efficient data management strategies for ease of addition and deletion of nodes and material points using advanced STL container, and nearest neighborhood communication for detection of neighbors and communication. Also, Linked Cell approach has been implemented to further reduce the computational efforts. Parallel performance analysis is investigated for challenging multiphysics applications like Taylor rod impact and serrated chip formation process. Adequate scalability of parallel implementation for these applications is reported.

Selective Laser Melting (SLM) is a laser powder bed fusion process that frequently comes into use for printing metal or ceramic structures. A positive connection between the layers can only be achieved by a remelt of the particles with the underlying layer. The key factor for successful printing is the control of the melt pool, which determines the shape and properties of the printed part. The parameters of the 3D printer must be set so that unwanted side effects can be excluded. The transition from the powder to the final printed part is a sequence of numerous physical processes that must be represented by the simulation. Suitable experimental tests are required to validate the models. Both aspects are presented in the first two sections. The differential equations are solved using SPH and the OTM method. The results of both schemes are compared and evaluated. The presented modeling approaches can also be used for Laser Metal Deposition or other laser powder bed fusion processes such as Electron Beam Melting or Selective Laser Sintering.

This paper presents a tool to estimate the stresses, and thus the expected forces, at a retaining wall by the installation of piles. The solution method is based on Smoothed Particle Hydrodynamics (SPH) and the soil is modeled using a simplified hypoplastic material law. In order to correctly compute the forces on the soil due to the contact between the pile and the soil, a formulation for imposing frictional boundary conditions using SPH is developed. Modeling the soil with the chosen hypoplastic approach also allows for tensile forces in the soil. However, these are not physical. Therefore, an alternative formulation is presented which directly eliminates unphysical tensile stresses in the cohesionless soil without any additional numerical parameters. The numerical code is firstly validated against benchmark problems. Then several test cases are simulated including monotonic and cyclic penetration of piles into the soil. A good agreement with the experimental observation is found. Additionally, the impact of pile driving in the presence of sheetpiles (retainers) is investigated to see how the pile driving can alter the applied forces on the sheet piles. The simulation of such complex geotechnical problems that involve large deformation, material nonlinearity, and moving boundary conditions demonstrates the applicability and versatility of the proposed numerical tool in this field.

The Peridynamic Petrov–Galerkin (PPG) method is a meshfree particle method based on the weak form of the peridynamic momentum equation. It can be applied to arbitrary constitutive laws from the classical continuum mechanics theory. With non-linear approximation functions the rank deficiency present in many nodally integrated discretization schemes is prevented. The consistency of trial functions is not sufficient for the convergence with irregular particle distributions. In this paper the consistency of the test space is examined and possible correction techniques are presented. The resulting variationally consistent PPG method is able to pass the patch test and to restore the optimal convergence rates. A correction of the test functions that preserves the linear trial function consistency allows the use of displacement–pressure–dilation formulations and exhibits stability and robustness for 3-D in the regime of non-linear elasticity. Besides, the direct nodal coupling with Finite Elements and the application of symmetry boundary conditions are enabled.

Photopolymerization is based on a chemical reaction. The material is initially in liquid to viscous form. As described in Chap. 2, different technologies for printing polymers exist. For the modeling, the process plays a minor role, since only the material parameters and the geometric structure differ. This chapter concentrates on the Additive Manufacturing process according to Stieghorst (2017). However, the modeling approach can also be applied to stereolithography or polyjet printing.

The simulation of Additive Manufacturing is based on coupled differential equations. Besides precise models for all phenomena, accurate solution schemes for differential equations are a prerequisite for realistic reproduction.

The Finite Element Method based on a Lagrangian description of the differential equation is mostly used to simulate the behavior of solids under loadings. With this scheme, a good approximation of the solution can be achieved, provided that the elements do not distort too much. However, this restriction limits the range of applications. To increase the flexibility of Galerkin methods, approaches are pursued which allow the determination of the test and trial function on an almost arbitrary distribution of nodes in the corresponding neighborhood.

Additive Manufacturing (AM) refers to the fabrication of products layer-by-layer from formless raw materials based on a data triple. This data triple consists of a 3D model of the part to be manufactured, the material recipe, and the process parameters.

Data-driven material models have many advantages over classical numerical approaches, such as the direct utilization of experimental data and the possibility to improve performance of predictions when additional data is available. One approach to develop a data-driven material model is to use machine learning tools. These can be trained offline to fit an observed material behaviour and then be applied in online applications. However, learning and predicting history dependent material models, such as plasticity, is still challenging. In this work, a machine learning based material modelling framework is proposed for both elasticity and plasticity. The machine learning based hyperelasticity model is developed with the Feed forward Neural Network (FNN) directly whereas the machine learning based plasticity model is developed by using of a novel method called Proper Orthogonal Decomposition Feed forward Neural Network (PODFNN). In order to account for the loading history, the accumulated absolute strain is proposed to be the history variable of the plasticity model. Additionally, the strain-stress sequence data for plasticity is collected from different loading-unloading paths based on the concept of sequence for plasticity. By means of the POD, the multi-dimensional stress sequence is decoupled leading to independent one dimensional coefficient sequences. In this case, the neural network with multiple output is replaced by multiple independent neural networks each possessing a one-dimensional output, which leads to less training time and better training performance. To apply the machine learning based material model in finite element analysis, the tangent matrix is derived by the automatic symbolic differentiation tool AceGen. The effectiveness and generalization of the presented models are investigated by a series of numerical examples using both 2D and 3D finite element analysis.

The Peridynamic Petrov–Galerkin (PPG) method is a meshfree approach based on the peridynamic integro-differential form of the momentum equation. The spurious oscillations in the common peridynamic correspondence formulation are investigated. They occur due to an inadmissible linearized mapping of the family deformation field. This leads to a generalized correspondence formulation, which contains the common formulation as a special case. It is based on the weak form of the peridynamic momentum equation. Test and trial function requirements are examined which ensure an exact imposition of Dirichlet and Neumann boundary conditions and Weighted Least Square (WLS) shape functions as well as Local Maximum Entropy (LME) approximants are utilized to examine the PPG Method. A consistent linearization is provided, which can also be used to speed up common implicit peridynamic correspondence codes. It is used in an implicit quasistatic framework to investigate the impact of different shape function combinations. Test cases show that low-energy modes can be prevented by the PPG Method and highlight the fast convergence and stability.