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Three recent breakthroughs due to AI in arts and science serve as motivation: An award winning digital image, protein folding, fast matrix multiplication. Many recent developments in artificial neural networks, particularly deep learning (DL), applied and relevant to computational mechanics (solid, fluids, finite-element technology) are reviewed in detail. Both hybrid and pure machine learning (ML) methods are discussed. Hybrid methods combine traditional PDE discretizations with ML methods either (1) to help model complex nonlinear constitutive relations, (2) to nonlinearly reduce the model order for efficient simulation (turbulence), or (3) to accelerate the simulation by predicting certain components in the traditional integration methods. Here, methods (1) and (2) relied on Long-Short-Term Memory (LSTM) architecture, with method (3) relying on convolutional neural networks. Pure ML methods to solve (nonlinear) PDEs are represented by Physics-Informed Neural network (PINN) methods, which could be combined with attention mechanism to address discontinuous solutions. Both LSTM and attention architectures, together with modern and generalized classic optimizers to include stochasticity for DL networks, are extensively reviewed. Kernel machines, including Gaussian processes, are provided to sufficient depth for more advanced works such as shallow networks with infinite width. Not only addressing experts, readers are assumed familiar with computational mechanics, but not with DL, whose concepts and applications are built up from the basics, aiming at bringing first-time learners quickly to the forefront of research. History and limitations of AI are recounted and discussed, with particular attention at pointing out misstatements or misconceptions of the classics, even in well-known references. Positioning and pointing control of a large-deformable beam is given as an example. [ABSTRACT FROM AUTHOR]

The present article, provides a novel mixed finite element formulation for elastoplasticity which is suitable for the discretization of thin‐walled structures using highly anisotropic volume elements. We present a thermodynamically consistent framework for the modeling of elastoplastic stress response, where dissipative effects are considered through a dissipation function instead of explicit flow rules. Along these lines, a mixed incremental principle, in which stresses are included as independent unknowns, is derived. We propose to employ tangential‐displacement normal‐normal‐stress (TDNNS) elements for a Galerkin discretization of the underlying variational problem. These elements were originally developed for linear elasticity, where it could be shown that they do not suffer from shear locking if the element's aspect ratio deteriorates. Excellent computational performance and accuracy of the proposed method are demonstrated in several benchmark problems. [ABSTRACT FROM AUTHOR]

This paper addresses the application of the continuum mechanics-based multiplicative decomposition for thermo-hyperelastic 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 stress-strain 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper presents a novel system‐level model order reduction scheme for flexible multibody simulation, namely the system‐level affine projection (SLAP). Contrary to existing system‐level model order reduction approaches for multibody systems simulation, this methodology allows to obtain a constant reduced order basis which can be obtained in a noninvasive fashion with respect to the original flexible multibody model. It is shown that this scheme enables an automatic joint constraint elimination which can be obtained at low computational cost through exploitation of the component level modes typically employed in flexible multibody simulation. The equations of motion are derived such that the computational cost of the resulting SLAP model is independent of the original model size. This approach results in a set of ordinary differential equations with a constant mass matrix and nonlinear internal forces. This structure makes the resulting model suitable for a range of estimation, control, and design applications. The proposed approach is validated numerically on a flexible four‐bar mechanism and shows good accuracy for a very low‐order SLAP model. [ABSTRACT FROM AUTHOR]

We are concerned with the mathematical modeling of the polarization process in ferroelectric media. We assume that this dissipative process is governed by two constitutive functions, which are the free energy function and the dissipation function. The dissipation function, which is closely connected to the dissipated energy, is usually non‐differentiable. Thus, a minimization condition for the overall energy includes the subdifferential of the dissipation function. This condition can also be formulated by way of a variational inequality in the unknown fields strain, dielectric displacement, remanent polarization and remanent strain. We analyze the mathematical well‐posedness of this problem. We provide an existence and uniqueness result for the time‐discrete update equation. Under stronger assumptions, we can prove existence of a solution to the time‐dependent variational inequality. To solve the discretized variational inequality, we use mixed finite elements, where mechanical displacement and dielectric displacement are unknowns, as well as polarization (and, if included in the model, remanent strain). It is then possible to satisfy Gauss' law of zero free charges exactly. We propose to regularize the dissipation function and solve for all unknowns at once in a single Newton iteration. We present numerical examples gained in the open source software package Netgen/NGSolve. [ABSTRACT FROM AUTHOR]

As compared to conventional mechanisms, compliant mechanisms exploit flexible deformation rather than rigid‐body motion of its components. The key advantage of jointless or monolithic structures lies in the absence of relative motion between the links. Besides the reduction in parts required to perform a task, compliant mechanisms typically show little friction, if any, and do not require lubrication to minimize wear. On the downside, design and synthesis of compliant mechanisms become much more involved than in rigid‐body linkages. In particular, optimization of compliant mechanisms relying on (sub‐)structures subjected to large flexible deformation is a challenging task, in which diverse aspects need to be considered. First and foremost, kinematic analysis of compliant mechanisms usually requires geometric non‐linearities to be accounted for. Further, limitations in actuation forces and torques impose constraints on the design. Depending on the application, the performance of compliant mechanisms may crucially depend on the natural frequencies and their change over the range of operation. In view of the diverse aspects, one typically has to deal with multi‐objective optimization problems in comparatively high‐dimensional parameter spaces. [ABSTRACT FROM AUTHOR]

We propose a new three-dimensional formulation based on the mixed tangential-displacement normal-normal-stress method for elasticity. In elastic tangential-displacement normal-normal-stress elements, the tangential component of the displacement field and the normal component of the stress vector are degrees of freedom and continuous across interelement interfaces. Tangential-displacement normal-normal-stress finite elements have been shown to be locking-free with respect to shear locking in thin elements, which makes them suitable for the discretization of laminates or macrofiber composites. In the current paper, we extend the formulation to piezoelectric materials by adding the electric potential as degree of freedom. [ABSTRACT FROM AUTHOR]

From wind turbines to capillary blood flow, problems of fluid-structure interaction occur across different scales of length and time. Owed to the multiple scales involved, the inherent complexity of coupled fluid and structural dynamics requires advanced numerical methods for the computational analysis. The proposed method relies on the coupling of modally reduced flexible multibody systems with fluids represented by the smoothed particle hydrodynamics method. A reduced-order basis is employed to describe small flexible deformation of a structural component relative to its large rigid body motion that is represented by a bodyfixed frame. An efficient evaluation of the surface deformation of the bodies involved is a vital ingredient for the coupling. Conventional approaches suffer from the drawback of flexible deformation being represented in a body-fixed local frame. The generalized component mode synthesis, in which the total displacement is interpolated, mitigates this computational limitation. The linear relation between generalized coordinates and the total displacement allows the coupling forces to be evaluated within the parallel fluid framework. Only the reduced set of equations of motion is solved on the solid side using an implicit time integration scheme with possibly large time-steps, whereas the particlebased fluid simulation relies on a fast explicit scheme. [ABSTRACT FROM AUTHOR]

The present article investigates a novel approach for the constitutive modeling of piezoelectric continua subjected to large deformation and strong electric fields, which is based on a multiplicative decomposition of the deformation gradient tensor, a concept frequently used in the description of inelastic processes. By splitting the total deformation into an inelastic and a purely elastic part, the notion of a stress-free intermediate configuration is introduced, which evolves from the referential undeformed state if an electric field is present. Considering large deformations, the field equations of electromechanics need to be written in a geometrically exact form, which requires the introduction of suitable electric field quantities that fit into the Lagrangian framework adopted in the proposed formulation. To give an example of constitutive equations, the behavior of piezoelectric materials according to Voigt’s linear theory is generalized to the nonlinear case. For this purpose, the piezoelectric part of the deformation gradient is identified by comparing the corresponding nonlinear strain measures with the electrically induced strains in the linear theory of piezoelectricity. The present article discusses the implications and the quantitative influence of the choice of different strain tensors on the material response within the discussed approach. [ABSTRACT FROM PUBLISHER]

Subject of the present work is the stability control of beam-type structures under a compressive force. Main scope of this article is to increase the critical buckling load by means of active feedback control. The compressive force is applied with the help of a cable which is fixed at the tip of the free end and passes through a point at the fixation of the beam. The structural displacement of the column is controlled by a constant gain feedback algorithm, with strain sensors used as input and piezoelectric actuators. A consistent theoretical formulation of this nonconservative system based on the Bernoulli–Euler beam theory is presented. The formulation takes into account the complete mechatronic system, including the influence of the controller with discrete strain measurements as well as distributed piezoelectric patches. The analytical results are compared to numerical simulations. The influence of the controller on the buckling load, the appearance of flutter instability and the limits of the controller’s operation range are highlighted. [ABSTRACT FROM PUBLISHER]

The present paper deals with vibrations of axially moving beams. Large displacements and finite deformations with respect to a co-moving reference configuration are studied. For that sake, an extension of Reissner's non-linear beam theory is used, which allows working with constitutive relations stated at the stress-strain level of continuum mechanics. We restrict to plane deformations and to the Bernoulli-Euler assumption for slender beams. In extension to formulations in the literature, we study a situation, in which a reference configuration with an unknown length is to be considered. Exemplary numerical results are presented for the case of a statically pre-deformed beam with an initially unknown reference configuration, which is set into an accelerated axial motion, where the length of the reference configuration however is kept fixed during the onset of vibrations. [ABSTRACT FROM AUTHOR]

Bringing into focus the design aspect of thin film electro‐active polymer actuators justifies the deployment of a structural mechanics framework. We propose a physically consistent constitutive model for such actuators, which is valid for plates and shells as material surfaces within a complete direct formulation. To this end, we use the principle of virtual work to deduce the general form of the constitutive law from an augmented Helmholtz free energy, as a function of the structural Green‐Lagrange type strain measures and of the material electric field, without the need of a‐priori assumptions concerning the state of strain and stress. Mechanical deformations of thin film devices– e.g. made of polyurethane– under the action of an external electric field, are caused by two different sources. On the one hand, the applied electric field causes a dielectric polarization of the polymer matrix, yielding to corresponding attractive electrostatic forces between the electroded surfaces resulting into a squeezing of the film. On the other hand, crystalline graft units with a certain natural, but arbitrarily directed polarization, embedded in between the polymer chains, have to align in the direction of the applied electric field such that a rotation of the whole crystal unit takes place. This rotation results in an additional macroscopic thickness squeeze– known as the electrostrictive effect. We treat both electromechanical coupling phenomena separately, where it turns out, that the electrostatic forces can be accounted for by an electrical contribution to the augmented free energy, whereas, the electrostrictive effect is taken into account in the elastic part of the augmented free energy by virtue of a hybrid multiplicative and additive decomposition of the plate/shell deformation measures. Benefiting from the structural mechanics formulation, we gain a lower– two‐dimensional– formulation, which provides a clear physical insight into the nature of the deformation process initiated by the external electric field. E.g. for the linearised problem, a comparison to the literature on thermoelastic plates and shells uncovers the action of the electric field as a combined source of self‐stresses. In order to solve particular problems, the constitutive relation of the geometrically and physically nonlinear formulation is implemented into our in‐house finite element code. The computed results, which were tested against results from the literature as well as against test problems of our previous work (where numerical integration of the three dimensional plate/shell augmented free energy through the thickness was employed), show a very good agreement. [ABSTRACT FROM AUTHOR]