This paper gives a review of integration algorithms for finite dimensional mechanical systems that are based on discrete variational principles. The variational technique gives a unified treatment of many symplectic schemes, including those of higher order, as well as a natural treatment of the discrete Noether theorem. The approach also allows us to include forces, dissipation and constraints in a natural way. Amongst the many specific schemes treated as examples, the Verlet, SHAKE, RATTLE, Newmark, and the symplectic partitioned Runge–Kutta schemes are presented.

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Discrete mechanics and variational integrators

SUMMARY We present an introduction to the spectral element method, which provides an innovative numerical approach to the calculation of synthetic seismograms in 3-D earth models. The method combines the £exibility of a ¢nite element method with the accuracy of a spectral method. One uses a weak formulation of the equations of motion, which are solved on a mesh of hexahedral elements that is adapted to the free surface and to the main internal discontinuities of the model. The wave¢eld on the elements is discretized using high-degree Lagrange interpolants, and integration over an element is accomplished based upon the Gauss^Lobatto^Legendre integration rule. This combination of discretization and integration results in a diagonal mass matrix, which greatly simpli¢es the algorithm. We illustrate the great potential of the method by comparing it to a discrete wavenumber/re£ectivity method for layer-cake models. Both body and surface waves are accurately represented, and the method can handle point force as well as moment tensor sources. For a model with very steep surface topography we successfully benchmark the method against an approximate boundary technique. For a homogeneous medium with strong attenuation we obtain excellent agreement with the analytical solution for a point force.

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Introduction to the spectral element method for three-dimensional seismic wave propagation

We present the spectral element method to simulate elastic-wave propagation in realistic geological structures involving complieated free-surface topography and material interfaces for two- and three-dimensional geometries. The spectral element method introduced here is a high-order variational method for the spatial approximation of elastic-wave equations. The mass matrix is diagonal by construction in this method, which drastically reduces the computational cost and allows an efficient parallel implementation. Absorbing boundary conditions are introduced in variational form to simulate unbounded physical domains. The time discretization is based on an energy-momentum conserving scheme that can be put into a classical explicit-implicit predictor/multi-corrector format. Long-term energy conservation and stability properties are illustrated as well as the efficiency of the absorbing conditions. The associated Courant condition behaves as Δ tC < O ( nel−1/nd N −2), with nel the number of elements, nd the spatial dimension, and N the polynomial order. In practice, a spatial sampling of approximately 5 points per wavelength is found to be very accurate when working with a polynomial degree of N = 8. The accuracy of the method is shown by comparing the spectral element solution to analytical solutions of the classical two-dimensional (2D) problems of Lamb and Garvin. The flexibility of the method is then illustrated by studying more realistic 2D models involving realistic geometries and complex free-boundary conditions. Very accurate modeling of Rayleigh-wave propagation, surface diffraction, and Rayleigh-to-body-wave mode conversion associated with the free-surface curvature are obtained at low computational cost. The method is shown to provide an efficient tool to study the diffraction of elastic waves by three-dimensional (3D) surface topographies and the associated local effects on strong ground motion. Complex amplification patterns, both in space and time, are shown to occur even for a gentle hill topography. Extension to a heterogeneous hill structure is considered. The efficient implementation on parallel distributed memory architectures will allow to perform real-time visualization and interactive physical investigations of 3D amplification phenomena for seismic risk assessment.

The spectral element method: An efficient tool to simulate the seismic response of 2D and 3D geological structures

SUMMARY We use a spectral-element method to simulate seismic wave propagation throughout the entire globe. The method is based upon a weak formulation of the equations of motion and combines the flexibility of a finite-element method with the accuracy of a global pseudospectral method. The finite-element mesh honours all first- and second-order discontinuities in the earth model. To maintain a relatively constant resolution throughout the model in terms of the number of grid points per wavelength, the size of the elements is increased with depth in a conforming fashion, thus retaining a diagonal mass matrix. In the Earth’s mantle and inner core we solve the wave equation in terms of displacement, whereas in the liquid outer core we use a formulation based upon a scalar potential. The three domains are matched at the inner core and core‐ mantle boundaries, honouring the continuity of traction and the normal component of velocity. The effects of attenuation and anisotropy are fully incorporated. The method is implemented on a parallel computer using a message passing technique. We benchmark spectral-element synthetic seismograms against normal-mode synthetics for a spherically symmetric reference model. The two methods are in excellent agreement for all body- and surface-wave arrivals with periods greater than about 20 s.

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Spectral-element simulations of global seismic wave propagation—I. Validation

A formulation and algorithmic treatment of static and dynamic plasticity at finite strains based on the multiplicative decomposition is presented which inherits all the features of the classical models of infinitesimal plasticity. The key computational implication is this: the closest-point-projection algorithm of any classical simple-surface or multi-surface model of infinitesimal plasticity carries over to the present finite deformation context without modification. In particular, the algorithmic elastoplastic tangent moduli of the infinitesimal theory remain unchanged. For the static problem, the proposed class of algorithms preserve exactly plastic volume changes if the yield criterion is pressure insensitive. For the dynamic problem, a class of time-stepping algorithms is presented which inherits exactly the conservation laws of total linear and angular momentum. The actual performance of the methodology is illustrated in a number of representative large scale static and dynamic simulations.

Algorithms for static and dynamic multiplicative plasticity that preserve the classical return mapping schemes of the infinitesimal theory

In the absence of external loads or in the presence of symmetries (i.e., translational and rotational invariance) the nonlinear dynamics of continuum systems preserves the total linear and the total angular momentum. Furthermore, under assumption met by all classical models, the internal dissipation in the system is non-negative. The goal of this work is the systematic design of conserving algorithms that preserve exactly the conservation laws of momentum and inherit the property of positive dissipation forany step-size. In particular, within the specific context of elastodynamics, a second order accurate algorithm is presented that exhibits exact conservation of both total (linear and angular) momentum and total energy. This scheme is shown to be amenable to a completely straightforward (Galerkin) finite element implementation and ideally suited for long-term/large-scale simulations. The excellent performance of the method relative to conventional time-integrators is conclusively demonstrated in numerical simulations exhibiting large strains coupled with a large overall rigid motion.

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The discrete energy-momentum method: conserving algorithms for nonlinear elastodynamics

It is shown that widely used implicit schemes, in particular the classical Newmark family of algorithms and its variants, generally fail to conserve total angular momentum for nonlinear Hamiltonian systems including classical rigid body dynamics, nonlinear elastodynamics, nonlinear rods and nonlinear shells. For linear Hamiltonian systems, it is well known that only the Crank-Nicholson scheme exactly preserves the total energy of the system. This conservation property is typically lost in the nonlinear regime. A general class of implicit time-stepping algorithms is presented which preserves exactly the conservation laws present in a general Hamiltonian system with symmetry, in particular the total angular momentum and the total energy. Remarkably, the actual implementation of this class of algorithms can be effectively accomplished by means of a simple two-step solution scheme which results in essentially no added computational cost relative to standard implicit methods. A complete analysis of these algorithms and a related class of schemes referred to as symplectic integrators is given. The good performance of the proposed methodology is demonstrated by means of three numerical examples which constitute representative model problems of nonlinear elastodynamics, nonlinear rods and nonlinear shells.

Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics

The long-term dynamic response of non-linear geometrically exact rods under-going finite extension, shear and bending, accompanied by large overall motions, is addressed in detail. The central objective is the design of unconditionally stable time-stepping algorithms which exactly preserve fundamental constants of the motion such as the total linear momentum, the total angular momentum and, for the Hamiltonian case, the total energy. This objective is accomplished in two steps. First, a class of algorithms is introduced which conserves linear and angular momentum. This result holds independently of the definition of the algorithmic stress resultants. Second, an algorithmic counterpart of the elastic constitutive equations is developed such that the law of conservation of total energy is exactly preserved. Conventional schemes exhibiting no numerical dissipation, symplectic algorithms in particular, are shown to lead to unstable solutions when the high frequencies are not resolved. Compared to conventional schemes there is little, if any, additional computational cost involved in the proposed class of energy–momentum methods. The excellent performance of the new algorithm in comparison to other standard schemes is demonstrated in several numerical simulations.

Non-linear dynamics of three-dimensional rods: Exact energy and momentum conserving algorithms

A numerical time-integration scheme for the dynamics of non-linear elastic shells is presented that simultaneously and independent of the time-step size inherits exactly the conservation laws of total linear, total angular momentum as well as total energy The proposed technique generalizes to non-linear shells recent work of the authors on non-linear elastodynamics and is ideally suited for long-term/large-scale simulations The algorithm is second-order accurate and can be immediately extended with no modification to a fourth-order accurate scheme The property of exact energy conservation induces a strong notion of non-linear numerical stability which manifests itself in actual simulations

N. Tarnow

Papers

The discrete energy-momentum method: conserving algorithms for nonlinear elastodynamics

Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics

Non-linear dynamics of three-dimensional rods: Exact energy and momentum conserving algorithms

A new energy and momentum conserving algorithm for the non‐linear dynamics of shells

How to render second order accurate time-stepping algorithms fourth order accurate while retaining the stability and conservation properties