Construction and analysis of higher order Galerkin variational integrators

We develop a class of C1‐continuous time integration methods that are applicable to conservative problems in elastodynamics. These methods are based on Hamilton's law of varying action. From the action of the continuous system we derive a spatially and temporally weak form of the governing equilibrium equations. This expression is first discretized in space, considering standard finite elements. The resulting system is then discretized in time, approximating the displacement by piecewise cubic Hermite shape functions. Within the time domain we thus achieve C1‐continuity for the displacement field and C0‐continuity for the velocity field. From the discrete virtual action we finally construct a class of one‐step schemes. These methods are examined both analytically and numerically. Here, we study both linear and nonlinear systems as well as inherently continuous and discrete structures. In the numerical examples we focus on one‐dimensional applications. The provided theory, however, is general and valid also for problems in 2D or 3D. We show that the most favorable candidate — denoted as p2‐scheme — converges with order four. Thus, especially if high accuracy of the numerical solution is required, this scheme can be more efficient than methods of lower order. It further exhibits, for linear simple problems, properties similar to variational integrators, such as symplecticity. While it remains to be investigated whether symplecticity holds for arbitrary systems, all our numerical results show an excellent long‐term energy behavior.

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“…This agrees with the discussion in Ref. [64] that integrators interpolating the displacement linearly in time can be at most of second order.…”

Section: Wwwzamm-journalorgsupporting

confidence: 93%

“…Following Ref. , the six schemes can be expressed in the form

[\begin{matrix} v_{n + 1} \\ ω u_{n + 1} \end{matrix}] = A [\begin{matrix} v_{n} \\ ω u_{n} \end{matrix}],

where

boldA

is the amplification matrix given in Appendix (supplementary material). The terms

u_{n}

and

v_{n}

denote the displacement and the velocity at time step

t_{n}

.…”

Section: Properties Of the Six Schemesmentioning

confidence: 99%

C¹‐continuous space‐time discretization based on Hamilton's law of varying action

2016

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“…Fig. 2 hints the extension of existing results for Hamiltons's principle [9] to the discrete D'Alembert principle, namely that the convergence rate cannot be increased by using higher order of quadrature than of approximation. It is interesting to note, that if the heat transfer is changed from Fourier's law to Green & Naghdi type II [3], the entropy is to be conserved analytically and so does the VI.…”

Section: Model Problemmentioning

confidence: 61%

Variational Integrators for Thermo‐Viscoelastic Discrete Systems

Kern

Romero

Groß

2015

Proc Appl Math and Mech

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Variational integrators are modern time-integration schemes based on a discretization of the underlying variational principle. In this paper, Hamilton's principle is approximated by an action sum, whose vanishing variation results in discrete Euler-Lagrange equations or, equivalently, in discrete evolution equations for the position and momentum. In order to include the viscous and thermal virtual work (mechanical and thermal virtual dissipation), Hamilton's principle is extended by D'Alembert terms, which account for the time evolution equation of the internal variable and Fourier's law.From this variational formulation, variational integrators using different orders of approximation of the state variables as well as of the quadrature of the action integral are constructed and compared. A thermo-viscoelastic double pendulum comprised of two discrete masses connected by generalized Maxwell elements, and subject to heat conduction between them serves as a discrete model problem. Higher Order Variational IntegratorsThis work is a further development of an existing variational integrator (VI) for discrete thermo-elastic systems [1]. Again, the common notation for VIs [2] is adopted as well as the concept of thermacy [3]. Thermacy is also known as "thermal displacement", since its time derivative corresponds to the temperature ϑ =α. This formulation results in a natural definition of the entropy as "thermal momentum" and entropy flux as "thermal force". The variational principle for thermo-mechanically coupled systems [4] uses the Lagrangian L = T − ψ. It is the foundation for the numerical discretization.The construction of a variational integrator (VI) goes in two steps. Firstly, the state variables are approximated in time. Using Lagrangian polynomials of degree p the approximation of one time step h = t k+1 − t k readsSecondly, a time-step-wise quadrature rule of the action-integral is applied, here using Gauss-type integration of order gThe same numerical integration is applied to the virtual work and rearranged over the time grid of the approximationdefining the discrete generalized forces F d n . Finally, evaluation of the discrete Lagrangian L d (q 0 , q 1 . . . , q p+1 ) and the discrete generalized forces according to D'Alembert's principle leads to the position-momentum form for the time stepping scheme Model ProblemThe thermoelastic double pendulum is a minimalistic model, commonly used [5,6] for testing numerical time integrators. For brevity only the enhancements of the previous stage [1] are detailed here. The first one is the replacement of the thermo-elastic springs by the generalized Maxwell elements (spring in parallel to a series connection of spring and dash-pot) as illustrated in Fig. 1. Thus the free energy of each of these two Maxwell elements takes the form [7]

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“…In these equations we first calculate implicitly the unknown values of the postion vector for every micro time step and the unknown LAGRANGE multiplier in Equation (43). In the "update" step we calculate also implicitly the unknown LAGRANGE multiplier and the linear momentum vector at the end of the micro time step, Equation (44) [6], [9].…”

Section: The Discrete Lagrangiañmentioning

confidence: 99%

Energy Conserving Time Integration Based on Galerkin-Variational Integrators With Constraints

Bartelt¹,

Groß²

2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

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Abstract. Often, Lagrange's formalism based on the Lagrangian is the preferred way in order to derive equations of motion for a mechanical system. Therefore, the first step is the formulation of the total kinetic and total potential energy. In this formalism, holonomic constraints can be taken into account by using the Lagrange multiplier method, and external and dissipative forces can be included variationally consistently by means of the Lagrange-D'Alembert principle. In this way, we are directly arrive at a weak formulation of the equations of motion without using a strong form and integration by parts. The variational time integration method is now based on the discretization of the action functional and the discrete variational calculus. This method leads to time stepping schemes called variational integrators.In this paper, we show that variational integrators based on higher-order shape functions and quadrature rules leads to a higher-order time approximation, which preserve the total linear and total angular momentum of the mechanical system. A further topic of the paper is a new implementation of the Lagrange multiplier method for holonomic constraints in the higher-order variational integrator, in order to compute bearing forces by means of Lagrange multipliers. Usually, variational integrators show an excellent long time behavior and bind the total energy error per time step, but are not able to preserve the total energy exactly. Therefore, we introduce a discrete gradient of the total potential energy in the variational integrator to preserve the total energy. We show different discrete gradients with different results for numerical stability. We can show that these time stepping schemes preserve the total linear momentum, total angular momentum as well as the total energy for every time step size. As numerical examples, we show motions of three-dimensional continua, which are discretized in space by the finite element method. We start with a free flying hyper-elastic rotor to show the preservation of total linear and total angular momentum in combination with higher-order accuracy in time. In the second example, we consider a hyper-elastic beam and apply the Lagrange multiplier method in order to calculate the bearing forces at fixed nodes. In the last example, the energy conservation is shown for different discrete gradients.

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Construction and analysis of higher order Galerkin variational integrators

Cited by 51 publications

References 29 publications

C¹‐continuous space‐time discretization based on Hamilton's law of varying action

C¹‐continuous space‐time discretization based on Hamilton's law of varying action

Variational Integrators for Thermo‐Viscoelastic Discrete Systems

Energy Conserving Time Integration Based on Galerkin-Variational Integrators With Constraints

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Construction and analysis of higher order Galerkin variational integrators

Cited by 51 publications

References 29 publications

C1‐continuous space‐time discretization based on Hamilton's law of varying action

C1‐continuous space‐time discretization based on Hamilton's law of varying action

Variational Integrators for Thermo‐Viscoelastic Discrete Systems

Energy Conserving Time Integration Based on Galerkin-Variational Integrators With Constraints

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Product

Resources

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C¹‐continuous space‐time discretization based on Hamilton's law of varying action

C¹‐continuous space‐time discretization based on Hamilton's law of varying action