In finite element analysis, mesh refinement is typically employed to improve accuracy by increasing spatial resolution in regions with steep solution gradients. This study presents an adaptive mesh refinement technique for dynamic fracture simulation based on the phase-field method. A multi-level node distance function is introduced using the phase-field variable to control mesh density. As damage evolves, the nodal spacing is adaptively refined according to the prescribed maximum spacing; whenever the computed distance exceeds this threshold, a new field node is introduced at the element center, ensuring the mesh evolves consistently with crack propagation. In addition, material damping effects are incorporated into the phase-field formulation to capture realistic dynamic fracture responses. Time integration is investigated using the Newmark scheme, the generalized-alpha method, and the backward implicit approach. The results indicate that, while the first two schemes are commonly applied, the backward implicit method provides superior stability in dynamic simulations. Furthermore, a staggered solution strategy is proposed in which both displacement and phase-field variables are iteratively and consistently updated within each solution step. The effectiveness of the proposed methodology is demonstrated through three numerical examples. The responses of elastic energy, kinetic energy, and energy dissipated by crack propagation are evaluated, together with the effects of material damping. The results confirm that the presented approach significantly improves computational efficiency while preserving accuracy and exhibits robust convergence behavior in highly dynamic fracture simulations.

Explicit time integration for immersed finite element discretizations severely suffers from the influence of poorly cut elements. In this contribution, we propose a generalized eigenvalue stabilization (GEVS) strategy for the element mass matrices of cut elements to cure their adverse impact on the critical time step size of the global system. We use spectral basis functions, specifically C0 continuous Lagrangian interpolation polynomials defined on Gauss-Lobatto-Legendre (GLL) points, which, in combination with its associated GLL quadrature rule, yield high-order convergent diagonal mass matrices for uncut elements. Moreover, considering cut elements, we combine the proposed GEVS approach with the finite cell method to guarantee definiteness of the system matrices. However, the proposed GEVS stabilization can directly be applied to other immersed boundary finite element methods. Numerical experiments demonstrate that the stabilization strategy achieves optimal convergence rates and recovers critical time step sizes of equivalent boundary-conforming discretizations. This also holds in the presence of weakly enforced Dirichlet boundary conditions using either Nitsche’s method or penalty formulations.

Since its introduction two decades ago, the Finite Cell Method (FCM) has proven highly effective for simulating problems with complex geometries using simple structured meshes. At its core, FCM embeds the computational domain implicitly by assigning material properties at the level of integration points, from which the stiffness matrix is constructed.This paper introduces the Discontinuous Finite Cell Method (DFCM)—an extension of FCM that treats boundary and interface conditions independently for each cell. Whereas FCM primarily focuses on element-level integration, DFCM emphasizes the consistent imposition of boundary conditions through flux coupling and Nitsche’s method. This enables the accurate treatment of interfaces without requiring inter-element connectivity and allows local tuning of the material scaling parameter α to accommodate different material conditions.As a result, DFCM is well suited for modeling both geometric and material discontinuities. The framework also supports non-uniform polynomial orders and local p-enrichment with minimal effort, allowing the resolution to be increased selectively near interfaces or regions of interest. In combination with cell decomposition and stitching techniques, DFCM can efficiently handle cells containing multiple material islands while maintaining numerical stability and a minimal number of degrees of freedom.Together, these features position DFCM as a flexible and robust framework for high-order analysis of complex, evolving domains.

In this work, the robustness of phase-field fracture simulations is increased by introducing a new hybrid approach to take advantages of both explicit and implicit solvers. Switching to an explicit solver at critical load cases enables us to define a threshold on the maximum number of staggered iterations per load step. Having this threshold would be very useful in complicated FCM problems since we cannot wait for a full convergence of the implicit solver at critical load steps. With the new hybrid method, the problem is implicit as long as the implicit solver can converge within a specified threshold; otherwise, the failed load step will be solved robustly using an explicit approach. The method can resolve the highly nonlinear part of the global load–displacement curve. The superiority of the approach is shown in different benchmark examples.

Simply applying the directional derivative either twice to the strain-energy density function (hyperelasticity) or once to the stress–strain state (Cauchy elasticity) does not lead to the symmetries of the fourth-order elasticity tensor specified in the literature. Moreover, there are many justifications and arguments for the desired symmetries, which are summarized in this contribution. Thus, a symmetrization operator has to be introduced to guarantee minor symmetry, since the symmetry of the strain tensor is frequently neglected but is needed to obtain results required for particular elasticity relations. A thorough investigation is provided for both Cauchy elasticity and hyperelasticity, and what conclusion can be drawn on by various assumptions.

Previous studies have demonstrated that filling a ship's double hull cavity with lightweight glass particles, like Poraver significantly enhances its ability to withstand collisions. The energy absorption capability of such particles can be improved by coating them in a fluidised spray granulation process. In order to find the extent of this improvement, a numerical model needs to be developed. For that purpose, the discrete element method (DEM) with the bonded particle method (BPM) is employed using the open-source code MUSEN. This allows the breakage of particles to be simulated using a cluster of particles bonded to each other. Since a single particle in the experiment is represented with an agglomerate of several smaller particles in the simulation, it significantly adds to the computational costs and limits the scale of the simulations which can be performed. This paper explores the different fidelities of a DEM–BPM numerical model with its advantages and disadvantages in its depiction of a coated particle breakage.

In recent years, particles have gained popularity as crash absorbers. To improve their mechanical properties, a coating layer can be applied. To predict the effect of this coating, a numerical model must be developed. For this purpose, the present study employs the discrete element method, extended by the bonded particle method, using both high- and low-fidelity approaches. In this framework, a single physical particle is modelled as a cluster or agglomerate of smaller particles bonded together. To identify the parameters involved, a sensitivity analysis is performed, followed by optimisation using the particle swarm algorithm, with calibration based on uniaxial single particle compression tests. Once an optimised parameter set is obtained, the models are validated against multi particle compression test results. The outcomes of this study demonstrate the potential of the proposed methodology for simulating large-scale compression tests of coated granular materials.

This paper addresses the prediction of fracture processes in complex geometries. In this approach, phase-field modeling is integrated into the finite cell method to have an efficient and robust simulation tool for brittle fracture. The non-negative moment fitting approach is employed as an integration scheme to reduce the number of integration points, which leads to shorter simulation times as well as lower memory requirements for history variables. Two 3D benchmark examples are provided to demonstrate the performance of the implemented approach compared to the standard quadrature based on the space-tree approach. Finally, the proposed method is applied to predict the sequence of strut fractures in brittle foams.

Staggered solution algorithms are a well-known alternative to monolithic approaches for solving strongly coupled multi-field problems. A coupling between the subproblems is achieved by exchanging coupling quantities between the solvers. Implicit coupling schemes are realized by letting the solvers solve each load or time step repeatedly until convergence up to a given coupling tolerance is achieved. In fluid-structure interaction as well as a variety of other problem classes, the equations governing the individual fields are nonlinear. Accordingly, each solver performs an inner iterative solution procedure that terminates once an inner tolerance is reached. The basic idea of this work builds on the possibility to adaptively adjust these inner tolerances based on carefully designed rules while preserving the black-box nature of the solvers. The resulting coupling scheme yields significant improvements in computational efficiency compared to classical schemes where the inner solver tolerances are held fixed. This is demonstrated in several numerical examples. The idea is tested in combination with state-of-the-art convergence acceleration schemes and can be realized within any staggered solution approach by only minor modifications to the participating solvers.

Immersed boundary methods have attracted substantial interest in the last decades due to their potential for computations involving complex geometries. Often these cannot be efficiently discretized using boundary-fitted finite elements. Immersed boundary methods provide a simple and fully automatic discretization based on Cartesian grids and tailored quadrature schemes that account for the geometric model. It can thus be described independently of the grid, e.g., by image data obtained from computed tomography scans. The drawback of such a discretization lies in the potentially small overlap between certain elements in the grid and the geometry. These badly cut elements with small physical support pose a particular challenge for nonlinear and/or dynamic simulations. In this work, we focus on problems in structural dynamics and acoustics and concentrate on solving them with explicit time-marching schemes. In this context, badly cut elements can lead to unfeasibly small critical time step sizes. We investigate the performance of implicit-explicit time marching schemes and two stabilization methods developed in previous works as potential remedies. While these have been studied before with regard to their effectiveness in increasing the critical time step size, their numerical efficiency has only been considered in terms of accuracy per degree of freedom. In this paper, we evaluate the computation time required for a given accuracy, which depends not only on the number of degrees of freedom but also on the selected spatial discretization, the sparsity patterns of the system matrices, and the employed time-marching scheme.

The numerical investigation of acoustic damping materials, such as foams, constitutes a valuable enhancement to experimental testing. Typically, such materials are modeled in a homogenized way in order to reduce the computational effort and to circumvent the need for a computational mesh that resolves the complex micro-structure. However, to gain detailed insight into the acoustic behavior, e.g., the transmittance of noise, such fully resolved models are mandatory. The meshing process can still be drastically simplified by using a fictitious domain approach. We propose the finite cell method, which combines the fictitious domain approach with high-order finite elements and resolves the complex geometry using special quadrature rules. In order to take into account the fluid-filled pores of a typical damping material, a coupled vibroacoustic problem needs to be solved. To this end, we construct two separate finite cell discretizations and prescribe coupling conditions at the interface in the usual manner. The only difference to a classical boundary fitted approach to vibroacoustics is that the fluid-solid interface is immersed into the respective discretization and does not correspond to the element boundaries. The proposed enhancement of the finite cell method for vibroacoustics is verified based on a comparison with commercial software and used within an exemplary application.

The numerical structural analysis of problems with complex geometries can be challenging, especially if standard finite elements are used. In contrast, immersed methods, such as the finite cell method, relieve the mesh generation such that simply shaped elements/cells can be used. Then, the domain boundary is considered during the numerical integration. In finite strain analysis, the elements/cells face large distortions, especially the cells that are intersected by the boundary. When the solution fails, remeshing can be applied to continue the simulation. This process is currently limited to geometries described by a triangulated surface. Therefore, the present work shows an alternative way of describing the deformed geometry by interpolating the displacement field. In this work, the inverse distance approach, and radial basis functions (RBF) with and without a constant extension are applied. It turns out that RBF with constant extension leads to the most robust results compared to the other methods. Moreover, different geometry description types are tested, and the present approach leads to promising results.

The present work is a comparative study of different data transfer techniques in the context of the finite cell method (FCM) in combination with remeshing for hyperelastic problems undergoing large deformations. The FCM is an immersed-boundary method that uses Cartesian grids for the discretization so as to avoid the generation of boundary conforming meshes. To overcome problems with heavily distorted meshes at large deformation states, we apply a remeshing procedure. During the remeshing, the data containing the deformation history has to be transferred between the meshes. In the present study, different methods are considered and compared: radial basis functions without and with polynomial extension, inverse distance weighting, and L_2 -projection applying the shape functions used in the FCM for the trial and test functions.

Finite element methods for displacement problems in hyperelasticity lead to systems of nonlinear equations. These equations are usually solved with Newton's method or a related method. The convergence of Newton's method depends heavily on the proximity of the initial guess to the numerical solution. Load step methods overcome problems with divergence by applying the load in increments, leading to a sequence of sub-problems with initial guesses closer to the numerical solution of each sub-problem, supporting the convergence. The downside of this approach is the high computational effort needed to solve the load steps. Based on a benchmark problem in high-order FEM, we extend traditional load step methods to a new approach exploiting the hierarchical basis used for the spatial discretization of the problem and saving up to 50% of computation time (vs. benchmark).

In this paper, a new boundary-conforming adaptive method for the numerical integration of trimmed elements is presented. The locations and weights of new integration points are determined based on special mapping formulations. The prerequisite of this technique is to describe the trimming curves/surfaces by parametric Bezier curves/surfaces within the parent element domain. The fitting error is under control, therefore the number of quadrature points for exact integration can be adjusted automatically based on the complexity of trimming curve and the corresponding integrand. The proposed method can be easily implemented into fictitious domain approaches. Different integration tasks as well as structural examples reveal that the proposed method delivers accurate and robust solutions for a wide variety of 2D/3D geometries with a rather low number of integration points.

This study aims to introduce a robust numerical approach to simulate complex small-scale mediums such as trabecular bone tissue in form of a cylindrical specimen taken from human vertebra. Consideration of previous related studies indicates that there are several challenges in utilizing standard finite element (FE) techniques for the analysis of such biomechanical structures. This is mainly due to their time-consuming procedure required for generating geometry conforming meshes. In this regard, the finite cell method (FCM) is an interesting alternative because it is based on the concept of the fictitious domain technique in which underlying meshes do not need to conform to the boundary of the domain. Since the considered trabecular bone tissue consists of a complex small-scale internal morphology, generating a FE mesh is rather complicated. So, the application of the FCM can be justified by overcoming the mentioned shortcomings of FE methods for this problem. Using FCM, it is possible to simulate the mentioned trabecular cylinder from vertebral body using higher order cells of regular shapes where the geometry is taken care of through the numerical integration. The input for the present numerical tool corresponding to the complex internal morphology of the proposed tissue is given by a high-resolution microCT scan. The outcome of the FCM will be compared to results obtained by mechanical testing of the specimen.

Wire arc additive manufacturing enables the production of components with high deposition rates and the incorporation of multiple materials. However, the manufactured components possess a wavy surface, which is a major difficulty when it comes to simulating the mechanical behavior of wire arc additively manufactured components and evaluation of experimental full-field measurements. In this work, the wavy surface of a thick-walled tube is measured with a portable 3D scanning technique first. Then, the surface contour is considered numerically using the finite cell method. There, hierarchic shape functions based on integrated Legendre polynomials are combined with a fictitious domain approach to simplify the discretization process. This enables a hierarchic p-refinement process to study the convergence of the reaction quantities and the surface strains under tension–torsion load. Throughout all considerations, uncertainties arising from multiple sources are assessed. This includes the material parameter identification, the geometry measurement, and the experimental analysis. When comparing experiment and numerical simulation, the in-plane surface strains are computed based on displacement data using radial basis functions as ansatz for global surface interpolation. It turns out that the finite cell method is a suitable numerical technique to consider the wavy surface encountered for additively manufactured components. The numerical results of the mechanical response of thick-walled tubes subjected to tension–torsion load demonstrate good agreement with real experimental data, particularly when employing higher-order polynomials. This agreement persists even under the consideration of the inherent uncertainties stemming from multiple sources, which are determined by Gaussian error propagation.

This Special Issue summarizes state-of-the-art contributions to computational solid and fluid mechanics.New computational approaches related to phase-field modeling, peridynamics and coupled problems are presented.This includes the statistical analysis of effective crack properties within a phase-field approach, the wave propagation and their reflections at material interfaces in the context of peridynamic computations, the simulation of wind turbine towers with liquid column dampers, the modeling of offshore systems composed of slender structures, an extension of the natural force density method to 3D problems and the topology optimization for composite materials.In addition, recent methods such as virtual elements, discrete elements, boundary elements, isogeometric analysis, and fictitious domain approaches are investigated.The following aspects are discussed: triangular virtual elements for Kirchhoff-Love shells, discrete elements for advanced manufacturing technologies, mathematical aspects of the collocation boundary element method for elasticity, and isogeometric analysis of flexible multibody systems.Moreover, the finite cell method for wire arc additive manufacturing and remeshing as well as eigenvalue stabilization for structures undergoing large elastoplastic deformations and the modeling of cracking in cortical bones based on a mesh fragmentation technique are discussed.Furthermore, this volume includes machine learning techniques and uncertainty quantification in the context of enhanced deep learning for vascular wall fracture analysis, PINN-based level-set formulation for the reconstruction of bubble dynamics, data-driven computational mechanics for constitutive modeling, interior point algorithms in single crystal plasticity and uncertainty quantification using time-separated stochastic mechanics.

Due to their ease of use and low cost, passive damping methods are a preferred mean for the reduction of noise in many engineering applications. This applies in particular to foam materials, which exhibit good acoustic and mechanical damping properties. The selection of suitable materials is usually carried out experimentally and can be very labor-intensive and time-consuming. For this reason, it is helpful to develop qualified numerical methodsthat can be used for material design and selection. Hence, foam materials must be characterized experimentally in order to enable a later comparison with vibroacoustic simulations. This contribution, therefore, aims at providing suitable parameters for the use in numerical analyses and their validation. Firstly, the microstructure of a foam specimen is captured by means of a CT scan. In addition to providing the geometry for the multi-physics simulations, this measurement is also used to determine characteristic foam features, for example, strut thickness and pore size distribution. In the second step, the frequency-dependent stiffness and damping properties of the material are determined by a special experimental setup utilizing an electrodynamic shaker. Here, the dynamic system is approximated as a single-mass oscillator, which is sufficiently accurate for low frequencies. These properties will later be used in the numerical model to evaluate different parameter identification approaches. In the third and last step of the experimental campaign, measurements with an impedance tube are conducted to obtain the coefficient of absorption. This material parameter is particularly suitable for comparing experiments and simulations. Finally, the correlation between the experimental results is examined to provide a deeper understanding of the foam materials.