Alexander Humer

We propose a novel mixed finite-element formulation for geometrically exact (Simo–Reissner) beams that introduces the moment vector as additional independent field. The specific mixed form allows for an element-local, discontinuous approximation of rotations, which is key to a simple and efficient discretization framework. The concept of discrete curvature provides a mathematically consistent treatment of rotation discontinuities. For linear constitutive laws, the mixed form is derived via a Legendre transform of the curvature-related strain energy. Objectivity is retained at the discrete level by interpolating relative rotations through a multiplicative split of the rotation field; path-independence is inherent to the total Lagrangian setting and verified numerically. Several benchmarks demonstrate optimal rates of convergence and accuracy, irrespective of the beam’s slenderness and order of approximation. Notably, the lowest-order element entirely avoids rotation interpolation by employing element-constant rotations only.

This contribution presents a homogenization approach for the computationally efficient modeling of two-dimensional solid bodies that interact through structured surfaces. Instead of resolving the detailed microscale geometry, the method introduces a finite-thickness surrogate layer that captures the essential mechanical response of the structured contact zone. The proposed formulation is based on the description of microscale quantities, such as (normal) gap and relative sliding velocity, which are transferred to the macroscale using internal volumetric variables. The elastic behavior of the surrogate layer is identified through a mean-field homogenization approach that accounts for the non-linear dependency of stress response on the relative slip. As a result, the anisotropic and history-dependent behavior of the interface can be captured within the model. The proposed method is validated using two benchmark problems: a vertical stack of two structured blocks and a knurled interference fit. In both cases, the finite-thickness layer accurately reproduces the mechanical response of the fully resolved model, i.e., stick–slip transitions, hysteresis, and partial interface separation. The good agreement with the reference solutions, combined with a significant reduction in computational cost, demonstrates the potential of the method for efficient multi-scale interface modeling.

The modeling and simulation of thin dielectric viscoelastic structures experiencing large deformations presents significant challenges. In this work, we introduce a shell formulation that captures the essential kinematic and constitutive features of these structures under transient electric loading. The formulation includes elastic and viscous membrane strains, curvature, as well as thickness deformation and the variation of the electric field through the thickness. A thermodynamically consistent model is developed and discretized using low-regularity shell elements within a variational framework. The proposed method is validated through several computational examples. The results demonstrate a high level of accuracy in capturing the transient large-strain behavior of dielectric elastomer shells even for very coarse discretizations.

The finite element simulation of thin viscoelastic membranes is a challenging problem. On the one hand, geometric dimensions of the membrane are adverse to classical finite element formulations, which require the thickness of the structure to be resolved. On the other hand, both material and geometric nonlinearities need to be considered in the large-strain regime. In the present paper, a Kirchhoff–Love shell formulation including thickness deformation is deduced for incompressible viscoelastic membranes at finite strain. For this formulation, a mixed shell element with low regularity is proposed to discretize a viscoelastic shell formulation based on a multiplicative split of the deformation gradient. The formulation is tested against full three-dimensional formulations in a variety of examples. Excellent accordance of results in quasi-static as well as dynamic simulations is observed.

In computational engineering, enhancing the simulation speed and efficiency is a perpetual goal. To fully take advantage of neural network techniques and hardware, we present the SLiding-window Initially-truncated Dynamic-response Estimator (SLIDE), a deep learning-based method designed to estimate output sequences of mechanical or multibody systems with primarily, but not exclusively, forced excitation. A key advantage of SLIDE is its ability to estimate the dynamic response of damped systems without requiring the full system state, making it particularly effective for flexible multibody systems. The method truncates the output window based on the decay of initial effects, such as damping, which is approximated by the complex eigenvalues of the systems linearized equations. In addition, a second neural network is trained to provide an error estimation, further enhancing the methods applicability. The method is applied to a diverse selection of systems, including the Duffing oscillator, a flexible slider-crank system, and an industrial 6R manipulator, mounted on a flexible socket. Our results demonstrate significant speedups from the simulation up to several millions, exceeding real-time performance substantially.

This paper addresses the application of the continuum mechanics-based multiplicative decomposition for thermohyperelastic materials by Lu and Pister to Reissner’s structural mechanics-based, geometrically exact theory for finite strain plane deformations of beams, which represents a geometrically consistent non-linear extension of the linear shear-deformable Timoshenko beam theory. First, the Lu-Pister multiplicative decomposition of the displacement gradient tensor is reviewed in a three-dimensional setting, and the importance of its main consequence is emphasized, i.e., the fact that isothermal experiments conducted over a range of constant reference temperatures are sufficient to identify constitutive material parameters in the stress-strain relations. We address various isothermal stress-strain relations for isotropic hyperelastic materials and their extensions to thermoelasticity. In particular, a model belonging to what is referred to as Simo-Pister class of material laws is used as an example to demonstrate the proposed procedure to extend isothermal stress-strain relations for isotropic hyperelastic materials to thermoelasticity. A certain drawback of Reissner’s structural-mechanics based theory in its original form is that constitutive relations are to be stipulated at the one-dimensional level, between stress resultants and generalized strains, so that the standardized small-scale material testing at the stress-strain level is not at disposal. In order to overcome this, we use a stress-strain based extension of the Reissner theory proposed by Gerstmayr and Irschik for the isothermal case, which we utilize here in the framework of the considered thermoelastic extension of the Simo-Pister stressstrain law. Consistent with the latter extension, we derive non-linear thermo-hyperelastic constitutive relations between stress-resultants and general strains. Special emphasis is given to linearizations and their consequences. A numerical example demonstrates the efficacy of the structural-mechanics approach in large-deformation problems.

This paper is concerned with modeling the polarization process in ferroelectric media. We develop a thermodynamically consistent model, based on phenomenological descriptions of free energy as well as switching and saturation conditions in form of inequalities. Thermodynamically consistent models naturally lead to variational formulations. We propose to use the concept of variational inequalities. We aim at combining the different phenomenological conditions into one variational inequality. In our formulation we use one Lagrange multiplier for each condition (the onset of domain switching and saturation), each satisfying Karush-Kuhn-Tucker conditions. An update for reversible and remanent quantities is then computed within one, in general nonlinear, iteration.

An energy-based model of the ferroelectric polarization process is presented in the current contribution. In an energy-based setting, dielectric displacement and strain (or displacement) are the primary independent unknowns. As an internal variable, the remanent polarization vector is chosen. The model is then governed by two constitutive functions: the free energy function and the dissipation function. Choices for both functions are given. As the dissipation function for rate-independent response is non-differentiable, it is proposed to regularize the problem. Then, a variational equation can be posed, which is subsequently discretized using conforming finite elements for each quantity. We point out which kind of continuity is needed for each field (displacement, dielectric displacement and remanent polarization) is necessary to obtain a conforming method, and provide corresponding finite elements. The elements are chosen such that Gauss’ law of zero charges is satisfied exactly. The discretized variational equations are solved for all unknowns at once in a single Newton iteration. We present numerical examples gained in the open source software package Netgen/NGSolve.

Smart materials respond to external stimuli, e.g., electric fields, which enables their use as sensors and actuators. The electromechanical coupling of the direct and converse piezoelectric effects, for instance, is used for both actuation and sensing in diverse engineering applications. The response of ferroelectric materials depends on their state of remanent polarization and the presence of an external electric field. To extend the operational range of sensors and actuators, an accurate understanding of the evolution of the material's state of polarization is imperative, which requires both physical and geometric nonlinearities to be taken into account. Moreover, polymeric smart materials like PVDF allow significantly larger deformation as compared to conventional piezoelectric ceramics. The electromechanical coupling in piezoelectric materials manifests in ferroelectric and ferroelastic hystereses, which are related to both reversible and irreversible processes. Focusing on the latter, we transfer phenomenological models for domain switching in ferroelectric materials to the geometrically nonlinear regime. For this purpose, we follow related concepts of geometrically nonlinear elastoplasticity, where the concept of a multiplicative decomposition of the deformation gradient plays a key role. Accordingly, an additional deformation path that describes the evolution of the poled state from the unpoled referential configuration is introduced. The constitutive response of the material to mechanical and electrical loads is discussed, and dissipative internal forces that drive the evolution of the remanent polarization are derived within a thermodynamical framework and the principle of maximum dissipation.