ABSTRACT At first glance, it appears that the contents of Karl Weierstraß's (1815–1897) lecture courses on the calculus of variations were made available to the public only in the seventh volume of his Mathematische Werke , and thus not until 30 years after his death. In reality, however, this material had already found its way into textbooks of other mathematicians, who duly acknowledged Weierstraß's contributions; notable examples include Adolf Kneser (1862–1930) with his Lehrbuch der Variationsrechnung of 1900 and Oskar Bolza (1857–1942) with his Lectures on the Calculus of Variations of 1904 and its translation, Vorlesungen über Variationsrechnung of 1909. Of particular importance for the dissemination of Weierstraß's contributions to the calculus of variations was the lecture course he delivered during the summer semester of 1879, which served as the basis of a worked‐out set of lecture notes published by the “Mathematischer Verein” at the University of Berlin. Furthermore, Weierstraß discussed the calculus of variations in his letters to Hermann Amandus Schwarz (1843–1921), particularly regarding his plans for a textbook on this subject.
ABSTRACT Electroplated coatings on metallic substrates have a wide range of practical applications ranging from corrosion or wear protection on parts, good conductivity on electrical contacts to coating systems for bipolar plates in fuel cells. The large field of applications in electrochemical engineering motivates several questions for scientific computation, such as the simulation of coating thickness distributions and alloy‐composition distributions on the cathode, leading to nonlinearly coupled PDEs with nonlinear boundary conditions. Although in recent years many advances have been made in the scientific computation of electrochemical systems for electroplating, important questions are still open. These include the integration of large reaction systems into the models and their numerical treatment as they have a major impact on the electroplating process and the resulting layer. The given contribution addresses the modelling of reactions in the galvanic cell as well as electrode reactions specifically and constructs a suitable methodology to simulate the modelled system to obtain reliable and meaningful simulation results. In particular, convenient strategies to decouple and simplify the resulting large system of PDEs for efficient and comparingly fast simulations are addressed.
ABSTRACT We estimate effective linear elastic properties from a single microstructure image using a sliding‐window subsampling strategy applied to fields obtained by computational micromechanics. The method yields both mean properties and variance estimates without requiring multiple realizations. We compare against a classical RVE study with 7500 microstructures and demonstrate comparable accuracy at significantly reduced computational cost. Variance reduction based on volume fraction matching is shown to significantly reduce both random and systematic errors, despite strong correlations between samples.
ABSTRACT This paper is intended to serve as a supplement to the work [ PAMM ‐ Proceedings in Applied Mathematics and Mechanics 24, no. 2 (2024): e202400004] to underline the significance of the Lode angle and Lode parameter—concepts attributed to Walter Lode—for the development of modern plasticity theory. Building on his original work, emphasis is placed on the interpretations of his contemporaries and subsequent generations, which finally have led to the relationships for these quantities that are in common use today. In contrast to the widespread recognition his work now enjoys, the person of Walter Lode had been largely forgotten over the years. This study aims to help bridge that gap by shedding some light on the life and work of one of the lesser‐known graduates of the Göttingen School. The fact that his life did not follow a particularly straightforward path is, of course, attributed to the special circumstances and upheavals of the era of the two world wars and the period in between.
ABSTRACT The previously introduced X‐FFT solver permits solving three‐dimensional linear elastic homogenization problems with high accuracy and efficiency. Due to the extended finite element discretization (X‐FEM), matrix‐inclusion problems may be approximated with the error convergence of interface‐conforming finite elements. By leveraging FFT‐based computational homogenization methods, efficient numerical computations are achieved. A crucial ingredient of the X‐FFT solver is its preconditioning strategy, which prevents ill‐conditioning by bounding the eigenvalues of the preconditioned system with some constant from above and below independently of the mesh spacing. This work presents an element‐based approach for computing explicit bounds on these constants and deduces a step size for the basic scheme using these bounds. Computational experiments analyze the sharpness of the derived step size, and we discuss its influence on other solution schemes.
ABSTRACT The paper deals with a drift‐diffusion model for semiconductor devices with Schottky contacts at all metal–semiconductor interfaces. The presented analytical investigations permit Boltzmann as well as Fermi–Dirac statistics for the charge‐carrier densities. We report on energy estimates as well as the existence and boundedness of weak solutions of the instationary van Roosbroeck system for this type of boundary conditions. Under additional assumptions, higher regularity of the solutions is obtained. Here, higher regularity results for scalar quasilinear parabolic PDEs are used. Exploiting the higher regularity of the solutions, the uniqueness of the solution is demonstrated.
ABSTRACT Rate‐independent single crystal plasticity requires robust solution methods due to the non‐uniqueness of slip system activity. The interior point method, which enforces the yield criterion via slack variables and a so‐called barrier term, has proven both robust and computationally efficient for this class of problems and has been applied to crystal plasticity in small‐strain and large‐strain frameworks. A drawback of the classic interior point method is its nested loop structure, which increases computational cost and limits warm‐starting capabilities. In this work, we propose a fixed barrier parameter formulation that reduces the solution procedure to a single Newton loop and exploits history information from prior load steps to construct a near‐optimal initial guess. Numerical studies at the material point level for the small‐strain framework demonstrate that convergence is achieved on average in 2–3 interior point Newton iterations, with a stress response comparable to interior point and visco‐plastic approaches from the literature.
ABSTRACT In this contribution, we present a model for shear‐flexible beams with a lattice‐like meso structure. The meso structure consists of a network of beams. The meso ‐ and macroscale are coupled by means of the Hill–Mandel condition. Due to the use of a structural model at the macroscale, the cross‐sectional dimensions of the representative volume element (RVE) are coupled to those of the macroscale beam model. Therefore, we use the term mesoscale instead of microscale. For the first time, we apply periodic boundary conditions to an RVE not only for the translational degrees of freedom, but also for the rotational degrees of freedom. This is done in a way that does not constrain axial strain, shear, torsion, or bending modes. New constraint equations are introduced to prevent rigid‐body translations and rotations, as well as a dependence of the homogenised quantities on the length of the RVE. The proposed method is evaluated using two benchmark tests, which show very good agreement between the numerical results and the reference solution.
ABSTRACT Polypropylene (PP) is a thermoplastic polymer with a wide range of applications in the automotive industry due to its desirable properties, including low weight, cost efficiency, and high recyclability. In service, these materials are exposed to both thermal and mechanical loading. This study investigates the thermoviscoelastic behavior of three PP grades in unconditioned and thermally preconditioned states by dynamic mechanical analysis (DMA) using temperature‐frequency sweeps. The resulting temperature‐ and frequency‐dependent material response is evaluated using the time‐temperature superposition principle (TTSP). The results indicate that the investigated PP grades exhibit thermorheological complex behavior. However, restricting the shifting procedure to the storage modulus still yields comparatively smooth master curves and therefore provides an engineering‐oriented approximation of the stiffness‐related response. The observed shift behavior further suggests that talc affects the absolute stiffness level and the effective relaxation behavior of PP. In addition, thermal preconditioning predominantly influences the high reduced‐frequency response, indicating an increased stiffness contribution within the short‐time and glassy regime. The resulting shift factors are fitted by Williams–Landel–Ferry (WLF) and Arrhenius functions, with the WLF function showing slightly better agreement. A linear generalized Maxwell model (GMM) provides a good approximation of the storage modulus master curves over a broad frequency range. From an automotive perspective, the increased storage modulus, that occurs with higher talc content and thermal preconditioning, indicates a higher dynamic stiffness under short‐term loading conditions. This may affect vibration behavior and local stress levels in PP components.
Thin‐walled composite structures are widely used in weight‐critical applications such as aircraft and spacecraft. However, ensuring the stability of such structures under various load cases remains a key challenge in their design and optimization. For omega‐stringer stiffened panels, the local buckling and postbuckling behavior are investigated using closed‐form analytical solutions. The stiffened panel under consideration consists of the skin plate with eccentrically attached stringer feet along the longitudinal sides of the panel, while the remaining part of the omega‐stringer is modeled by corresponding elastically restrained edges. The computational model is based on energy methods and approximates the postbuckling behavior near the bifurcation point using a simplified plate model. To evaluate the new analysis method, a comparison with the finite element analysis is being drawn. Compared to numerical methods, the present model reduces the computational effort, which is particularly advantageous in the design phase.
We consider systems of delay differential equations (DDEs), including a single delay and a quadratic right‐hand side. In a system, parameters are replaced by random variables to perform an uncertainty quantification. Thus the solution of the DDEs becomes a random process, which can be represented by a series of the generalised polynomial chaos. We investigate the application of the stochastic Galerkin method and stochastic collocation techniques to compute unknown coefficient functions of the series. Furthermore, existence and local stability of stationary solutions are discussed for each type of method. We present results of numerical computations in two illustrative examples: a logistic equation and an epidemiological model.
This paper investigates the effects of cube‐shaped neighborhoods in peridynamic theory as an alternative to the traditional spherical neighborhoods. We examine how different neighborhood geometries influence the behavior of various peridynamic formulations, including bond‐based models, state‐based formulations, and correspondence methods. The study reveals that cube‐shaped neighborhoods introduce significant anisotropic effects in standard peridynamic formulations due to directional bias in force calculations. A generalized bond constant is derived for the bond‐based model to maintain consistency with classical continuum mechanics when using cubic neighborhoods. Through numerical examples, including wave propagation and crack growth scenarios, we demonstrate that while cube‐shaped neighborhoods cause undesirable anisotropic behavior in traditional peridynamic models, correspondence‐based formulations remain unaffected due to their averaged deformation gradient approach. The results provide important insights for the selection of neighborhood shapes in peridynamic simulations and highlight the robustness of correspondence formulations against geometric variations in discretization choices.
Quantum computing utilizes the underlying principles of quantum mechanics to perform computations with unmatched performance capabilities. Rather than using classical bits, it operates on qubits, which can exist in superposition and entangled states. This enables the solution of problems that are considered intractable for classical computers. However, since qubits are realized by physical systems such as the spins of electrons, they are highly sensitive to environmental disturbances and hardware imperfections. To achieve reliable scaling and practical application in the future, addressing these errors is of utmost importance. Different classes of errors exist, such as coherent and incoherent errors, caused by imperfections in quantum operations or the decoherence of quantum states. They are inherently different, as they arise from either a lack of precision or intrinsic randomness. Current literature struggles to provide a unified framework that models both types of errors simultaneously. In this paper, an approach based on possibility theory—a theory of imprecise probabilities—is presented to model quantum uncertainty. Possibility theory is particularly useful for systems affected by both epistemic and aleatoric uncertainty, that is, uncertainty due to limited knowledge and uncertainty due to inherent randomness, respectively. By exploring noisy quantum algorithms within a possibilistic framework, different statements about robustness can be derived without requiring prior assumptions about the underlying noise model. Moreover, a possibilistic model enables the derivation of sampling criteria for guaranteed statistical performance and provides insight into the number of measurements required—an important consideration, given that such resources are costly in practice.
This work delves into the advancement of topology optimization techniques for buckling structures that are subject to size limitations. Conventional density‐based methods are prone to yielding intricate, fine‐scale geometries that are challenging to fabricate. To circumvent this challenge, this work imposes constraints on the density distribution and utilizes aggregation techniques to consolidate local volume constraints into a unified global restriction. This approach ensures a more uniform distribution of material throughout the structure, thereby reducing material clustering, avoiding excessively thin layers, and enhancing both the manufacturability and structural performance. Numerical examples are employed to optimize the topology of buckling structures with varying size constraint parameters. The optimization outcomes demonstrate that the proposed methods are capable of effectively achieving optimal topologies that exhibit enhanced stability and strength while adhering to the specified size constraints.
Traditional pedagogical setups in laboratories are often outdated and do not provide the necessary support for effective learning. Teaching fluid mechanics poses additional significant challenges, particularly due to the need to visualize invisible properties, like velocity vector fields (velocity) or scalar pressure fields in time‐dependent 3D spaces. In this context, the application constructive alignment (CA) promises a more holistic approach to achieving better learning success. Here, the intended learning outcomes (ILOs), the teaching‐learning activities (TLAs), and the learning outcome monitoring (LOM), are taken as an iterative process and the three parts have to be perfectly aligned. To rate the ILOs, a cognitive taxonomy should be used. In an engineering context, we prefer the structure of observed learning outcome (SOLO) taxonomy as it helps to clarify, whether an ILO addresses surface understanding, or deep understanding. This work features a work in progress modular augmented reality (AR) mobile application called the AR FLOW for interactive fluid mechanic experiments. This application allows students to access worksheets via QR codes and explore various levels of learning. Furthermore, they can later access the laboratories that are set up in an interactive level system that helps them discover all the features of a level. The easy implementation of this application into existing lectures and courses is supported by accompanying teaching materials based on the principles of CA, featuring clear ILOs, and assessment tasks (ATs) rephrased according to SOLO taxonomy.
In aerospace transportation and propulsion systems, shock‐induced flow separation has strong detrimental effects on the aerodynamic behavior and performance. To alleviate these effects, separation control is necessary. A commonly pursued approach uses vortex generators (VG) of different design to increase the momentum transfer within the boundary layer and thus make it less prone to separation. Different mechanical vortex generators, valued in aerospace engineering for their robustness and simplicity, as well as the more flexible and less drag‐penalty prone air‐jet vortex generators (AJVGs) have been studied. A large number of parameters influence the control effectiveness of these devices, amongst them geometrical parameters, flow parameters, and the array arrangement of multiple devices. The latter aspect is particularly relevant for AJVGs, which are small enough to allow for inter‐device spacings that enable interactions between the turbulent structures induced by neighboring AJVGs, and where the intensity of these interactions strongly influence the control effect. With this great number of control parameters, vortex generators may easily encounter off‐design conditions in engineering applications, where operating conditions vary and are not as clean as in a laboratory. An overview on the relevant control parameters is provided, both for mechanical vortex generators including microramps and microvanes and for air‐jet vortex generators. Then, influences of off‐design conditions are discussed on the basis of results from recent experimental and numerical studies. A joint analysis of this rich data set allows an in‐depth interpretation of observed flow phenomena and control effects. Finally, we assess the application potential of the investigated control devices.
Crystal plasticity simulations offer insights into the anisotropic deformation of polycrystalline materials such as metals and alloys. However, rate‐independent crystal plasticity models encounter the Taylor ambiguity, where the active slip systems and plastic slip magnitudes are not uniquely defined, posing well‐known numerical challenges [1]. Interior point methods, which smooth the problem via a barrier term, have recently emerged as a promising strategy for both small‐strain [2, 3] and large‐strain [4, 5] crystal plasticity frameworks. This contribution presents a finite‐strain crystal plasticity model based on an interior point [2, 3]. Distinctly from existing large‐strain IPM implementations, our approach incorporates sequential updates for slip system rotations, which are held constant during intermediate interior point iterations, significantly enhancing algorithmic robustness. The method is evaluated at the material point level for several test cases, and its predictions are shown to be consistent with established results from various crystal plasticity algorithms in the literature, such as those based on an augmented Lagrange method.
This work presents a stable time‐domain boundary element method for the acoustic wave equation in three‐dimensional unbounded domains. Other formulations of time‐domain boundary element methods based on retarded potential operators are known to exhibit stability issues, which often hinder their use in industrial contexts. We have investigated the stability properties of a Galerkin first‐kind boundary integral formulation for sound emission problems, where well‐posedness can be established in both the continuous and the discrete setting. Numerical experiments confirm the accuracy and convergence of the method. We assess long‐time stability through extensive simulations focusing on fine temporal resolutions and large time ranges. The proposed formulation is compared with two alternative approaches used in practice: a space‐time single‐layer potential approach and a semi‐discretized collocation method.
Chemo‐mechanically coupled phenomena such as stress‐driven diffusion and diffusion‐induced stresses are of high interest, for example, in battery materials and metals. In this work, a chemo‐mechanically fully coupled multiphase‐field model for a multicomponent system is derived and validated with a sharp interface solution. Ensuring mechanical compatibility, the model accounts for balance equations on singular surfaces and the Hadamard jump conditions. The models' capability to address stress‐driven diffusion and diffusion‐induced stresses is demonstrated through the presentation of an illustrative diffusion example.
This study presents a numerical investigation of passive scalar mixing in homogeneous isotropic turbulence (HIT). Different volumetric forcing schemes have been used in the literature, but the side effects are rarely discussed, either because these are assumed irrelevant or because it is too costly to conduct such an analysis with a high-fidelity model. In this study, we have used One-Dimensional Turbulence (ODT) model to compare forcing schemes at low Reynolds numbers (upto Re λ = 70 ${\rm Re}_{\lambda } = 70$ ). Our analysis reveals critical flaws in the linear forcing model when applied to ODT. While both schemes exhibit spectral deviations from direct numerical simulation (DNS), the stochastic forcing scheme demonstrates superior dynamic fidelity, better capturing the turbulent energy cascade. In contrast, the linear forcing scheme suffers from a non-physical energy deficit at large scales and is approximately 10 times more computationally expensive. These artefacts directly impact scalar mixing: The stochastic scheme produces classic, multi-scale intermittency, whereas linear forcing generates extreme gradients confined only at the dissipative scales. These results demonstrate that the choice of forcing is a critical modelling decision in ODT, leading to fundamentally different model-dependent artefacts in both turbulence dynamics and scalar mixing statistics, at least in low Reynolds number regimes.
