Accurate modeling of ultrasound wave propagation is essential for high-fidelity simulation and imaging in ultrasonic testing. A primary challenge lies in characterizing the excitation source, particularly for transducers with large apertures relative to the acoustic wavelengths. In such cases, non-uniform excitation and spatial interference significantly affect the resulting radiation patterns. This paper proposes a distributed source inversion strategy to reconstruct an effective spatio-temporal transducer model that reproduces experimentally measured wavefields. The reconstructed source model captures aperture-dependent phase and amplitude variations without requiring detailed knowledge of the transducer structure. The approach is validated using directivity measurements on an aluminum half-cylinder, where simulations incorporating the reconstructed source model show close agreement with experimental directivity patterns and waveform shapes. Finally, synthetic studies on reverse time migration and full-waveform inversion demonstrate that accurate transducer modeling is critical for the success of simulation-based imaging and inversion workflows and significantly improves reconstruction quality.

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.

Phased array ultrasound is a widely used technique in non-destructive testing. Using piezoelectric elements as both sources and receivers provides a significant gain in information and enables more accurate defect detection. The process is called full matrix capture when all source-receiver combinations are used. The total focusing method (TFM) exploits such datasets and relies on a delay and sum algorithm to sum up the signals on a pixel grid. However, TFM only uses the first arriving p-waves, making it challenging to size complex-shaped defects. By contrast, more advanced methods such as reverse time migration (RTM) and full waveform inversion (FWI) use full waveforms to reconstruct defects. Both methods compare measured signals with ultrasound simulations. While RTM identifies defects by convolving forward and backward wave fields once, FWI iteratively updates material models to reconstruct the actual distributions of material properties. This study compares TFM, RTM, and FWI for six specimens featuring circular defects or Y-shaped notches. The reconstructed results are first evaluated qualitatively by visual assessment, then different defect segmentation approaches are studied, and finally, they are evaluated quantitatively using metrics such as AUPRC, AUROC, and F1-score. The results show that a two-step FWI framework performs best in most of the six samples studied, both qualitatively and quantitatively.

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.

Efficient solution strategies for wave propagation problems with complex geometries are essential in many engineering fields. Immersed boundary methods simplify mesh generation by embedding the domain of interest into an extended domain that is easy to mesh, introducing the challenge of dealing with cells that intersect the domain boundary. We consider the finite cell method that extends the weak form from the physical into the extended domain, multiplied by a small value, to stabilize badly cut cells. Combined with explicit time integration methods, the presence of finite cells with very little support in the physical domain results in tiny critical time step sizes. While the finite cell stabilization limits how small the critical time step size can become, this limit is still restrictive for many applications. Explicit transient analyses commonly use the spectral element method due to its natural way of obtaining diagonal mass matrices through nodal lumping. The resulting method is very efficient since nodal lumping renders the solution of the equation systems trivial while not reducing the accuracy or the critical time step size. The combination of the spectral element and finite cell methods is called the spectral cell method. Unfortunately, a direct application of nodal lumping in the spectral cell method is impossible due to the special quadrature necessary to treat the discontinuous integrand inside the cut cells. The existing approaches to lump the mass matrices of cut cells significantly reduce the accuracy of the approximation. We analyze an implicit-explicit (IMEX) time integration method to exploit the advantages of the nodal lumping scheme for uncut cells on one side and the unconditional stability of implicit time integration schemes for cut cells on the other. In this hybrid, immersed Newmark IMEX approach, we use explicit second-order central differences to integrate the uncut degrees of freedom that lead to a diagonal block in the mass matrix and an implicit trapezoidal Newmark method to integrate the remaining degrees of freedom (those supported by at least one cut cell). The immersed Newmark IMEX approach preserves the high-order convergence rates and the geometric flexibility of the finite cell method while retaining the efficiency of the nodal lumping scheme for uncut cells without compromising the critical time step size. We analyze a simple system of spring-coupled masses to highlight some of the essential characteristics of Newmark IMEX time integration. We then solve the scalar wave equation with immersed boundaries on two- and three-dimensional examples with significant geometric complexity to show that our approach is more efficient than state-of-the-art time integration schemes when comparing accuracy and runtime. While we focus on the finite and spectral cell methods, we expect other immersed methods to benefit equally from Newmark IMEX time integration.

Full Waveform Inversion (FWI) is a successful and well-established inverse method for reconstructing material models from measured wave signals. In the field of seismic exploration, FWI has proven particularly successful in the reconstruction of smoothly varying material deviations. In contrast, non-destructive testing (NDT) often requires the detection and specification of sharp defects in a specimen. If the contrast between materials is low, FWI can be successfully applied to these problems as well. However, so far the method is not fully suitable to image defects such as voids, which are characterized by a high contrast in the material parameters. In this paper, we introduce a dimensionless scaling function $γ$ to model voids in the forward and inverse scalar wave equation problem. Depending on which material parameters this function $γ$ scales, different modeling approaches are presented, leading to three formulations of mono-parameter FWI and one formulation of two-parameter FWI. The resulting problems are solved by first-order optimization, where the gradient is computed by an ajdoint state method. The corresponding Fréchet kernels are derived for each approach and the associated minimization is performed using an L-BFGS algorithm. A comparison between the different approaches shows that scaling the density with $γ$ is most promising for parameterizing voids in the forward and inverse problem. Finally, in order to consider arbitrary complex geometries known a priori, this approach is combined with an immersed boundary method, the finite cell method (FCM).

Full waveform inversion (FWI) is an iterative identification process that serves to minimize the misfit of model-based simulated and experimentally measured wave field data. Its goal is to identify a field of parameters for a given physical object. For many years, FWI is very successful in seismic imaging to deduce velocity models of the earth or of local geophysical exploration areas. FWI has also been successfully applied in various other fields, including non-destructive testing (NDT) and biomedical imaging. The inverse optimization process of FWI relies on forward and backward solutions of the (elastic or acoustic) wave equation, as well as on efficient computations of adequate optimization directions. Many approaches employ (low order) finite element or finite difference methods, using parameterized material fields whose resolution is chosen in relation to the elements or nodes of the discretized wave field. In our previous paper (Burchner et al., 2023), we investigated the potential of using the finite cell method (FCM) as the wave field solver. The FCM offers the advantage that highly complex geometric models can be incorporated easily. Furthermore, we demonstrated that the identification of the model's density outperforms that of the velocity - particularly in cases where unknown voids characterized by homogeneous Neumann boundary conditions need to be detected. The paper at hand extends this previous study in the following aspects: The isogeometric finite cell analysis (IGA-FCM) - a combination of isogeometric analysis (IGA) and FCM - is applied as the wave field solver, with the advantage that the polynomial degree and subsequently also the sampling frequency of the wave field can be increased quite easily. Since the inversion efficiency strongly depends on the accuracy of the forward and backward wave field solutions and of the gradients of the functional, consistent, and lumped mass matrix discretization are compared. The resolution of the grid describing the unknown material density - thus allowing to identify voids in a physical object - is then decoupled from the knot span grid. Finally, we propose an adaptive multi-resolution algorithm that locally refines the material grid using an image processing-based refinement indicator. The developed inversion framework allows fast and memory-efficient wave simulations and object identification. While we study the general behavior of the proposed approach using 2D benchmark problems, a final 3D problem shows that it can also be used to identify void regions in geometrically complex spatial structures. (c) 2023 Elsevier B.V. All rights reserved.

The advancements in additive manufacturing (AM) technology have allowed for the production of geometrically complex parts with customizable designs. This versatility benefits large-scale space-frame structures, as the individual design of each structural node can be tailored to meet specific mechanical and other functional requirements. To this end, however, the design and analysis of such space-frames with distinct structural nodes needs to be highly automated. A critical aspect in this context is automated integration of the local 3D features into the 1D large-scale models. In the present work, a two-scale modeling approach is developed to improve the design and linear-elastic analysis of space frames with complex additively manufactured nodes. The mechanical characteristics of the 3D nodes are numerically reduced through an automated dimensional reduction process based on the Finite Cell Method (FCM) and substructuring. The reduced stiffness quantities are assembled in the large-scale 1D model which, in turn, enables efficient structural analysis. The response of the 1D model is passed on to the local model, enabling fully resolved 3D linear-elastic analysis. The proposed approach is numerically verified on a simplified beam example. Furthermore, the workflow is demonstrated on a tree canopy structure with additively manufactured nodes with bolted connections. The form of the large-scale structure is found based on the Combinatorial Equilibrium Modeling framework, and the different designs of the local structural nodes are based on generative exploration of the design space. It is demonstrated that the proposed methodology effectively automates the design and analysis of space-frame structures with complex, distinct structural nodes.

Ultrasonic methods have great potential applications to detect and characterize defects in multi-layered bonded composites. However, it remains challenging to quantitatively reconstruct defects, such as disbonds, that influence the integrity of adhesive bonds and seriously reduce the strength of assemblies. In this work, an ultrasonic method based on the supervised fully con-volutional network (FCN) is proposed to quantitatively reconstruct high contrast defects hidden in multi-layered bonded composites. In the training process of this method, an FCN establishes a non-linear mapping from measured ultrasonic data to the corresponding longitudinal wave (L-wave) velocity models of multi-layered bonded composites. In the predicting process, the network obtained from the training process is used to directly reconstruct the L-wave velocity models from the new measured ultrasonic data of adhesively bonded composites. The presented FCN-based inversion method can automatically extract useful features in multi-layered compos-ites. Although this method is computationally expensive in the training process, the prediction itself in the online phase takes only seconds. The numerical and experimental results show that the FCN-based ultrasonic inversion method is capable of accurately reconstructing ultrasonic L -wave velocity models of the high contrast defects, which has great potential for online detection of adhesively bonded composites.

Proximal humerus impacted fractures are of clinical concern in the elderly population. Prediction of such fractures by CT-based finite element methods encounters several major obstacles such as heterogeneous mechanical properties and fracture due to compressive strains. We herein propose to investigate a variation of the phase field method (PFM) embedded into the finite cell method (FCM) to simulate impacted humeral fractures in fresh frozen human humeri. The force-strain response, failure loads and the fracture path are compared to experimental observations for validation purposes. The PFM (by means of the regularization parameter ℓ0) is first calibrated by one experiment and thereafter used for the prediction of the mechanical response of two other human fresh frozen humeri. All humeri are fractured at the surgical neck and strains are monitored by Digital Image Correlation (DIC). Experimental strains in the elastic regime are reproduced with good agreement (R2=0.726), similarly to the validated finite element method (Dahan et al., 2022). The failure pattern and fracture evolution at the surgical neck predicted by the PFM mimic extremely well the experimental observations for all three humeri. The maximum relative error in the computed failure loads is 3.8%. To the best of our knowledge this is the first method that can predict well the experimental compressive failure pattern as well as the force-strain relationship in proximal humerus fractures.

Combining the additive manufacturing (AM) process of extrusion with lightweight concrete, mono-material but multi-functional elements with an internal cellular structure can be created to achieve good thermal performance of a wall at low resource consumption. The aim of this paper is to analyze and optimize the actual thermal performance of such a component. A sensitivity analysis and a parametric optimization were conducted based on a mathematical description of heat transfer in cellular structures. To investigate the thermal performance, 2D and 3D heat transfer simulations were used and validated by heat flux measurements on an existing prototype. A geometric optimization led to a further reduction of the U-value by up to 24%, reaching 0.58 W/m2 K. The ratio of solid material to air inside the cells (relative density) was identified as the main driver, in addition to cell diameter, cell height, and cell wall thickness. The comparison of analytical and numerical results showed high correspondence with deviations of 3–10%, and for the experimental results 25%. These remaining deviations can be traced back to simplifications of the theoretical models and discrepancies between as designed and as built. The presented approach provides a good basis for optimizing the thermal design of complex AM components by investigating practical thermal problems with the help of 2D and 3D simulations, and thus offers a great potential for further applications.

The finite element method is classically based on nodal Lagrange basis functions defined on conforming meshes. In this context, total reaction forces are commonly computed from the so-called “nodal forces”, yielding higher accuracy and convergence rates than reactions obtained from the differentiated primal solution (“direct” method). The finite cell method and isogeometric analysis promise to improve the interoperability of computer-aided design and computer-aided engineering, enabling a direct approach to the numerical simulation of trimmed geometries. However, body-unfitted meshes preclude the use of classic nodal reaction algorithms. This work shows that the direct method can perform particularly poorly for immersed methods. Instead, conservative reactions can be obtained from equilibrium expressions given by the weak problem formulation, yielding superior accuracy and convergence rates typical of nodal reactions. This approach is also extended to non-interpolatory basis functions, such as the (truncated) hierarchical B-splines.

We present an efficient algorithmic framework for constructing multi-level hp-bases that uses a data-oriented approach that easily extends to any number of dimensions and provides a natural framework for performance-optimized implementations. We only operate on the bounding faces of finite elements without considering their lower-dimensional topological features and demonstrate the potential of the presented methods using a newly written open-source library. First, we analyze a Fichera corner and show that the framework does not increase runtime and memory consumption when compared against the classical p-version of the finite element method. Then, we compute a transient example with dynamic refinement and derefinement, where we also obtain the expected convergence rates and excellent performance in computing time and memory usage.

This work presents a new ultrasonic imaging framework for non-destructive evaluation of components with vertical or steeply dipping surfaces and demonstrates its ability of accurately characterizing multiple defects hidden in the interior of the component based on a limited coverage of ultrasonic linear phased array. Central to the framework is a point cloud-based elastic reverse time migration (PC-based ERTM) method. First, a surface reconstruction is derived from the point cloud provided through photos of an object from multiple views by bundle adjustment. Second, by taking the surface reconstruction as a geometric background estimate for elastic reverse time migration, the algorithm considers information of multiple scattering and mode conversions as well as multiple wave reflections from the component’s bottom and aims at detecting internal defects. The effectiveness and accuracy of the PC-based ERTM approach is examined by experiments with multiple defects in extruded aluminum specimens with vertical surfaces. Experimental results show that notches and side-drilled holes in components can be reconstructed accurately.

We apply the recently developed least squares stabilized symmetric Nitsche method for enforcement of Dirichlet boundary conditions to the finite cell method. The least squares stabilized Nitsche method in combination with finite cell stabilization leads to a symmetric positive definite stiffness matrix and relies only on elementwise stabilization, which does not lead to additional fill in. We prove a priori error estimates and bounds on the condition numbers.

The direct numerical simulation of metal additive manufacturing processes such as laser powder bed fusion is challenging due to the vast differences in spatial and temporal scales. Classical approaches based on locally refined finite elements combined with time-stepping schemes can only address the spatial multi-scale nature and provide only limited scaling potential for massively parallel computations. We address these shortcomings in a space-time Galerkin framework where the finite element interpolation also includes the temporal direction. In this setting, we construct four-dimensional meshes that are locally refined towards the laser spot and allow for varying temporal accuracy depending on the position in space. By splitting the mesh into conforming time slabs, we recover a stepwise solution to solve the space-time problem locally in time at this slab; additionally, we can choose time-slab sizes significantly larger than classical time-stepping schemes. As a result, we believe this setting to be well suited for large-scale parallelization. In our work, we use a continuous Galerkin-Petrov formulation of the nonlinear heat equation with an apparent heat capacity model to account for the phase change. We validate our approach by computing the AMB2018-02 benchmark, where we obtain an excellent agreement with the measured melt pool shape. Using the same setup, we demonstrate the performance potential of our approach by hatching a square area with a laser path length of about one meter.

Process-induced defects are the leading cause of discrepancies between as-designed and as-manufactured additive manufacturing (AM) product behavior. Especially for metal lattices, the variations in the printed geometry cannot be neglected. Therefore, the evaluation of the influence of microstructural variability on their mechanical behavior is crucial for the quality assessment of the produced structures. Commonly, the as-manufactured geometry can be obtained by computed tomography (CT). However, to incorporate all process-induced defects into the numerical analysis is often computationally demanding. Thus, commonly this task is limited to a predefined set of considered variations, such as strut size or strut diameter. In this work, a CT-based binary random field is proposed to generate statistically equivalent geometries of periodic metal lattices. The proposed random field model in combination with the Finite Cell Method (FCM), an immersed boundary method, allows to efficiently evaluate the influence of the underlying microstructure on the variability of the mechanical behavior of AM products. Numerical analysis of two lattices manufactured at different scales shows an excellent agreement with experimental data. Furthermore, it provides a unique insight into the effects of the process on the occurring geometrical variations and final mechanical behavior.