Energy-momentum Schemes Research Articles

Advanced discretization techniques for hyperelastic physics-augmented neural networks

In the present work, advanced spatial and temporal discretization techniques are tailored to hyperelastic physics-augmented neural networks, i.e., neural network based constitutive models which fulfill all relevant mechanical conditions of hyperelasticity by construction. The framework takes into account the structure of neural network-based constitutive models, in particular, that their derivatives are more complex compared to analytical models. The proposed framework allows for convenient mixed Hu–Washizu like finite element formulations applicable to nearly incompressible material behavior. The key feature of this work is a tailored energy–momentum scheme for time discretization, which allows for energy and momentum preserving dynamical simulations. Both the mixed formulation and the energy–momentum discretization are applied in finite element analysis. For this, a hyperelastic physics-augmented neural network model is calibrated to data generated with an analytical potential. In all finite element simulations, the proposed discretization techniques show excellent performance. All of this demonstrates that, from a formal point of view, neural networks are essentially mathematical functions. As such, they can be applied in numerical methods as straightforwardly as analytical constitutive models. Nevertheless, their special structure suggests to tailor advanced discretization methods, to arrive at compact mathematical formulations and convenient implementations.

Structure-preserving integrators based on a new variational principle for constrained mechanical systems

A new variational principle for mechanical systems subject to holonomic constraints is presented. The newly proposed GGL principle is closely related to the often used Gear-Gupta-Leimkuhler (GGL) stabilization of the differential–algebraic equations governing the motion of constrained mechanical systems. The GGL variational principle relies on an extension of the Livens principle (sometimes also referred to as Hamilton–Pontryagin principle) to mechanical systems subject to holonomic constraints. In contrast to the original GGL stabilization, the new approach facilitates the design of structure-preserving integrators. In particular, new variational integrators are presented, which result from the direct discretization of the GGL variational principle. These variational integrators are symplectic and conserve momentum maps in the case of systems with symmetry. In addition to that, a new energy–momentum scheme is developed, which results from the discretization of the Euler–Lagrange equations pertaining to the GGL variational principle. The numerical properties of the newly devised schemes are investigated in representative examples of constrained mechanical systems.

On the design of non-singular, energy-momentum consistent integrators for nonlinear dynamics using energy splitting and perturbation techniques

This work proposes a suite of numerical techniques to facilitate the design of structure-preserving integrators for nonlinear dynamics, with particular emphasis on many-body dynamics. The celebrated LaBudde-Greenspan integrator and various energy-momentum schemes adopt a difference quotient formula in their algorithmic force definitions, which suffers from numerical instability as the denominator gets close to zero. There is a need to develop structure-preserving integrators without invoking the quotient formula. In this work, the potential energy of a Hamiltonian system is split into two parts, and specially developed quadrature rules are applied separately to them. The resulting integrators can be regarded as classical ones perturbed with first- or second-order terms, and the energy split guarantees the dissipative nature in the numerical residual. In the meantime, the conservation of invariants is respected in the design. A complete analysis of the proposed integrators is given, with representative numerical examples provided to demonstrate their performance. They can be used either independently as energy-decaying and momentum-conserving schemes for nonlinear problems or as an alternate option with a conserving integrator, such as the LaBudde-Greenspan integrator, when the numerical instability in the difference quotient is detected.

Incorporation of thermal effects into the energy‐momentum consistent elastic model for fiber‐bending stiffness in composites

AbstractOwing to the wide acceptance of fiber reinforced composites in the lightweight structure industry, the need to study its thermo‐elastic behavior in dynamical multi‐body systems still remains an open question. Standard continuum models, by virtue of first‐order gradient of deformation, cannot capture curvature effects and hence appropriate extensions of the standard modeling techniques are indispensable to model the accurate physical stiffness. The fiber‐bending curvature are captured by introducing a higher‐order gradient of the deformation as an independent field in the standard continuum. This work explores in detail, a further extension of this continuum to understand the thermal effects on the bending stiffness of fiber bundle in fiber‐reinforced composites. Our proposed model incorporates the temperature of the composite and its bidirectional dependency with the higher‐order gradients of the deformation into the extended mechanical continuum model. The implementation of the model is pursued in the context of multi‐field mixed finite element method and dynamic variational principle. The principle of virtual power enables our extended non‐isothermal continuum to derive an energy‐momentum scheme of higher order. The resulting time‐integrator exhibits physically consistent locking‐ free numerical behavior in dynamical simulations. Our novel approach is illustrated by simple transient dynamic simulations with a hyperelastic, transversely isotropic, polyconvex material behavior. We investigate the conservation of total energy and total momentum in the framework of thermo‐elasticity and study the spatial and temporal convergence for long‐term dynamic simulations with coarser grids and increased accurate numerical solutions.

A thermodynamically consistent time integration scheme for non-linear thermo-electro-mechanics

The aim of this paper is the design of a new one-step implicit and thermodynamically consistent Energy–Momentum (EM) preserving time integration scheme for the simulation of thermo-electro-elastic processes undergoing large deformations. The time integration scheme takes advantage of the notion of polyconvexity and of a new tensor cross product algebra. These two ingredients are shown to be crucial for the design of so-called discrete derivatives fundamental for the calculation of the second Piola–Kirchhoff stress tensor, the entropy and the electric field. In particular, the exploitation of polyconvexity and the tensor cross product, enable the derivation of comparatively simple formulas for the discrete derivatives. This is in sharp contrast to much more elaborate discrete derivatives which are one of the main downsides of classical EM time integration schemes. The newly proposed scheme inherits the advantages of EM schemes recently published in the context of thermo-elasticity and electro-mechanics, whilst extending to the more generic case of nonlinear thermo-electro-mechanics. Furthermore, the manuscript delves into suitable convexity/concavity restrictions that thermo-electro-mechanical strain energy functions must comply with in order to yield physically and mathematically admissible solutions. Finally, a series of numerical examples will be presented in order to demonstrate robustness and numerical stability properties of the new EM scheme.

A new energy–momentum time integration scheme for non-linear thermo-mechanics

The aim of this paper is the design a new one-step implicit and thermodynamically consistent Energy–Momentum (EM) preserving time integration scheme for the simulation of thermo-elastic processes undergoing large deformations and temperature fields. Following Bonet et al. (2020), we consider well-posed constitutive models for the entire range of deformations and temperature. In that regard, the consideration of polyconvexity inspired constitutive models and a new tensor cross product algebra are shown to be crucial in order to derive the so-called discrete derivatives, fundamental for the construction of the algorithmic derived variables, namely the second Piola–Kirchoff stress tensor and the entropy (or the absolute temperature). The proposed scheme inherits the advantages of the EM scheme recently published by Franke et al. (2018), whilst resulting in a simpler scheme from the implementation standpoint. A series of numerical examples will be presented in order to demonstrate the robustness and applicability of the new EM scheme. Although the examples presented will make use of a temperature-based version of the EM scheme (using the Helmholtz free energy as the thermodynamical potential and the temperature as the thermodynamical state variable), we also include in an Appendix an entropy-based analogue EM scheme (using the internal energy as the thermodynamical potential and the entropy as the thermodynamical state variable).

Non-isothermal energy–momentum time integrations with drilling degrees of freedom of composites with viscoelastic fiber bundles and curvature–twist stiffness

3D fiber-reinforced composites demand for a special simulation technique, because they consist of fiber bundles. Therefore, in the corresponding representative volume element of these metamaterials, secondary effects as a micro inertia and a curvature–twist stiffness have to bear in mind. The latter increases the strength-to-weight ratio of thin-walled lightweight structures due to a separate twisting and bending stiffness of the fiber bundles. In this paper, these secondary effects are introduced in a continuum formulation by means of independent drilling degrees of freedom. The resulting non-isothermal constrained micropolar continuum is derived by a principle of virtual power, which simultaneously generates in the discrete setting a mixed B-bar method and a Galerkin-based energy–momentum scheme of higher order. This work also takes into account viscoelastic material behavior in the fiber bundles, which arises from a mixture of organic and inorganic fibers. Here, the viscous evolution equation is solved elementwise by using a mixed field as viscous internal variable. Representative numerical examples demonstrate the inelastic material behavior, the effect of micro inertia on the physical properties of the continuum as well as on its space and time discretization and, finally, the twisting and bending stiffness of the fiber bundles. Further, non-standard boundary conditions are applied in the dynamic simulations performed by higher order energy–momentum schemes.

A general framework for the thermodynamically consistent time integration of open nonlinear thermoelastic systems

AbstractThis work deals with Energy‐Momentum‐Entropy (EME) consistent methods of open thermoelastic systems. While Energy‐Momentum (EM) preserving integrators are well‐known for conservative mechanical systems, Romero introduced in [1] the class of thermodynamically consistent (TC) integrators for coupled thermomechanical systems, which further respect symmetries of the underlying coupled system and are capable of conserving associated momentum maps and can therefore also be termed EME schemes in analogy to EM schemes for conservative systems. As mathematical framework for the geometric structure of the non‐equilibrium thermodynamics the GENERIC (General Equation for Non‐Equilibrium Reversible‐Irreversible Coupling) framework, originally introduced for complex fluids in [2], is used. Since the GENERIC framework does not depend on a specific choice of the thermodynamical state variables [3], we explore the structure of the GENERIC framework using the entropy density (see e.g. [4]), the absolute temperature (see e.g. [5]) and further the internal energy density as thermodynamical state variable. Applying the notion of a discrete gradient in the sense of [6] leads to an EME integrator. As boundary conditions rely on the specific choice of the thermodynamical state variable we extend the GENERIC framework to be suitable for open systems following the procedure in [7].

Energy and symmetry-preserving formulation of nonlinear constraints and potential forces in multibody dynamics

Practical multibody models are typically composed by a set of bodies (rigid or deformable) linked by joints, represented by constraint equations, and in many cases are subject to potential forces. Thus, a proper formulation of these constraints and forces is an essential aspect in the numerical analysis of their dynamics. On the other hand, geometric integrators are particular time-stepping schemes that have been successfully employed during the last decades in many applications, including multibody systems. One category is the so-called energy-momentum (EM) schemes that exhibit excellent stability and physical accuracy, conserving the discrete energy in conservative systems and preserving possible symmetries related with the conservation of linear and angular momenta. The use of the discrete derivative concept greatly systematizes their formulation, but its particular expression is not unique. This paper discusses the properties of several discrete derivative formulas proposed in the literature for models possessing finite-dimensional and linear configuration spaces, showing that not all of them lead to proper EM schemes. Some formulas, when applied to constraints or potentials endowed with certain symmetries (associated with the conservation of linear and angular momenta, found in many common practical systems), may produce numerical results that conserves the energy but violates the symmetries, introducing numerical instabilities and/or producing unphysical motions. This fact, surprisingly overlooked by many authors, is carefully analyzed and illustrated with several numerical experiments related with the dynamics of representative multibody systems.

Variational-based locking-free energy–momentum schemes of higher-order for thermo-viscoelastic fiber-reinforced continua

Locking-free finite elements and energy–momentum time integration schemes are two of the best-known algorithmic improvements of finite element methods. Both are developed since the middle of the eighties of the last century, but usually independently from each other. Therefore, a smart interface between both algorithms is rarely a development goal of the researcher. In this paper, we present such a smart interface, namely the Hu–Washizuprocedure applied to the principle of virtual power. By using of the resulting mixed variational principle, we avoid locking in a spatial finite element discretization of non-isothermal inelastic fiber-reinforced materials, and obtain a family of corresponding higher-order accurate energy–momentum schemes. Thereby, we consider volumetric locking in the matrix material and line locking in the fibers. We show that this reduction of locking in the energy–momentum schemes leads to an increase of the maximum time step size, such that the efficiency of the time integration is improved in the sense that less CPU time is required. This could be shown by using an automatic time step size control with the iteration number of the applied Newton–Raphson scheme as target function. As numerical examples, we consider slender fiber-reinforced structures as a turbine rotor and a lightweight beam consisting of fiber-reinforced trusses. Here, we simulate different combinations of mechanical and thermal Dirichlet and Neumann boundary conditions.

Variational-based higher-order accurate energy–momentum schemes for thermo-viscoelastic fiber-reinforced continua

In this paper, new higher-order accurate energy–momentum schemes are presented, which rely on a discrete mixed principle of virtual power. The schemes are designed for simulating an uni-directional fiber-reinforced continuum, considered as transversally isotropic nonlinear continuum with independent material behavior in and normal to fiber direction. The matrix material is considered as an isotropic thermo-viscoelastic material and the fibers behave thermo-elastic. Hence, the model takes into account an independent conduction of heat according to Duhamel’s law with a transversally isotropic conductivity tensor as well as an independent heat expansion and heat capacity of the matrix and the fibers. The energy–momentum schemes preserve each balance law of the continuous problem also in the discrete setting, independent of the chosen time step size and the prescribed Neumann and Dirichlet boundary conditions. Therefore, the implemented time step size control with the iteration number as target function does not influence the structure-preservation of the schemes. The balance laws are also preserved together with different time scales in the mechanical, thermal and viscous time evolution. By calculating the generalized reactions on the boundary, numerical examples show energy–momentum consistent dynamic simulations of different transient Dirichlet and Neumann boundary conditions.

A mixed variational framework for the design of energy–momentum schemes inspired by the structure of polyconvex stored energy functions

A new approach to the design of energy–momentum (EM) consistent algorithms for nonlinear elastodynamics is proposed. The underlying mixed variational formulation is motivated by the structure of polyconvex stored energy functions and benefits from the notion of a tensor cross product for second-order tensors. The structure-preserving discretization in time of the mixed variational formulation yields an EM consistent semi-discrete formulation. The semi-discrete formulation offers several options for the discretization in space. In the special case of a purely displacement-based method a new form of the algorithmic stress formula is obtained. Several numerical examples are presented to evaluate the performance of the newly developed schemes.

Variational-based energy–momentum schemes of higher-order for elastic fiber-reinforced continua

This paper contributes to the improvement of recently published higher-order accurate energy–momentum schemes for an anisotropic hyperelastic material formulation based on the concept of structural tensors. These Galerkin-based energy–momentum schemes are developed by means of time finite elements with the focus on improving numerical stability and robustness in the presence of stiffness and large rotations. They show advantages over conventional time stepping schemes in combination with anisotropic materials formulated with polyconvex strain energy density functions. In this paper, higher-order energy–momentum schemes are developed by using a differential variational principle in order to remedy the drawbacks of these Galerkin-based schemes, namely a variationally inconsistent time approximation of the strains in the higher-order case, no coordinate free superimposed stress tensor and an unseparated time approximation of the fibers and the matrix material. These new structure-preserving time integrators preserve all conservation laws of a free flying hyperelastic anisotropic continuum, namely the total linear and the total angular momentum conservation as well as the total energy conservation. Numerical examples demonstrate dynamic behavior of anisotropic continua, discrete conservation properties of flying continua and the effect of higher-order accuracy subject to transient loads.

Energy‐momentum conserving discretization of mixed shell elements for large deformation problems

AbstractIn the present paper we consider structure‐preserving integration methods in the context of mixed finite elements. The used low‐order mixed finite elements typically exhibit improved coarse mesh accuracy. On the other hand energy‐momentum (EM) consistent time‐stepping schemes have been developed in the realm of nonlinear structural dynamics to enhance the numerical stability properties. EM schemes typically exhibit superior robustness and thus offer the possibility to use large time steps while still producing physically meaningful results. Accordingly, combining mixed finite element discretizations in space with EM consistent discretizations in time shows great promise for the design of numerical methods with superior coarse mesh accuracy in space and time. Starting with a general Hu‐Washizu‐type variational formulation we develop a second‐order accurate structure‐preserving integration scheme. The present approach is applicable to a large number of mixed finite element formulations. As sample application we deal with a specific mixed shell element. Numerical examples dealing with large deformations will show the improved coarse mesh accuracy in space and time of the advocated approach. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Isogeometric analysis and domain decomposition methods

In the present work we use the mortar finite element method for the coupling of nonconforming discretized sub-domains in the framework of nonlinear elasticity. The mortar method has been shown to preserve optimal convergence rates (see Laursen (2002) [25] for details) and is variationally consistent. We show that the method can be applied to isogeometric analysis with little effort, once the framework of NURBS based shape functions has been implemented. Furthermore, a specific coordinate augmentation technique allows the design of an energy–momentum scheme for the constrained mechanical system under consideration. The excellent performance of the redesigned mortar method as well as the energy–momentum scheme is illustrated in representative numerical examples.

Optimal Control of Multibody Dynamics with Contact

AbstractThis work discusses two different structure preserving integrators in the framework of optimal control simulations with contact. The first one is a variational integrator, based on the constrained version of the Lagrange‐D'Alembert. The resulting scheme preserves the symplecticity and the momentum maps of the simulated multibody dynamics. The second integrator is an energy momentum scheme and it is based on the augmented Hamiltonian equations, which are discretised using the discrete derivative in [2]. Both integrators are applied to simulate the optimal control of compass gait, for which the contact between the foot and the ground is modelled as perfectly plastic contact. The second example represents a monopedal jumper and it is used to examine the dynamical behaviour of the perfectly elastic and perfectly plastic contact formulation. The resulting differential algebraic equations (DAEs) are solved by the aforementioned symplectic momentum method. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Transient 3d contact problems—NTS method: mixed methods and conserving integration

The present work deals with a new formulation for transient large deformation contact problems. It is well known, that one-step implicit time integration schemes for highly non-linear systems fail to conserve the total energy of the system. To deal with this drawback, a mixed method is newly proposed in conjunction with the concept of a discrete gradient. In particular, we reformulate the well known and widely-used node-to-segment methods and establish an energy-momentum scheme. The advocated approach ensures robustness and enhanced numerical stability, demonstrated in several three-dimensional applications of the proposed algorithm.

A comparison of structure-preserving integrators for discrete thermoelastic systems

This paper contains a comparison of three recently proposed structure-preserving time-stepping schemes for nonlinear thermomechanical systems. These schemes can be considered as extension to coupled thermoelastic problems of well-established energy–momentum schemes for nonlinear elastodynamics. The present comparison is performed in the context of a finite-dimensional model problem for coupled thermomechanical systems: the thermoelastic double pendulum. It is shown that, similar to their purely mechanical ancestors, structure-preserving integrators for coupled thermoelasticity in general exhibit superior numerical stability and robustness properties.

Variational Integrators and Energy-Momentum Schemes for Flexible Multibody Dynamics

This work contains a comparison between variational integrators and energy-momentum schemes for flexible multibody dynamics. In this connection, a specific “rotationless” formulation of flexible multibody dynamics is employed. Flexible components such as continuum bodies and geometrically exact beams and shells are discretized in space by using nonlinear finite element methods. The motion of the resulting discrete systems are governed by a uniform set of differential-algebraic equations (DAEs). This makes possible the application and comparison of previously developed structure-preserving methods for the numerical integration of the DAEs. In particular, we apply a specific variational integrator and an energy-momentum scheme. The performance of both integrators is assessed in the context of three representative numerical examples.

Transient three‐dimensional domain decomposition problems: Frame‐indifferent mortar constraints and conserving integration

AbstractThe present work deals with transient large‐deformation domain decomposition problems. The tying of dissimilar meshed grids is performed by applying the mortar method. In this connection, a reformulation of the original linear mortar constraints is proposed, which retains frame‐indifference for arbitrary discretizations of the interface. Furthermore, a specific coordinate augmentation technique is proposed to make possible the design of an energy–momentum scheme. Numerical examples demonstrate the robustness and enhanced numerical stability of the newly developed energy–momentum scheme for three‐dimensional problems. Copyright © 2009 John Wiley & Sons, Ltd.