Time finite element methods for structural dynamics

Time finite element methods are developed for the equations of structural dynamics. The approach employs the time-discontinuous Galerkin method and incorporates stabilizing terms having least-squares form. These enable a general convergence theorem to be proved in a norm stronger than the energy norm. Results are presented from finite difference analyses of the time-discontinuous Galerkin and least-squares methods with various temporal interpolations and commonly used finite difference methods for structural dynamics. These results show that, for particular interpolations, the time finite element method exhibits improved accuracy and stability.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/50098/1/1620330206_ftp.pd

Discontinuity-capturing operators for elastodynamics

Finite element methods are presented which include discontinuity-capturing operators to accurately solve elastodynamics problems exhibiting sharp gradients in their solution, e.g., wave propagation. The formulation involves finite element discretization of the temporal domain as well as the usual finite element discretization of the spatial domain. Linear least-squares terms are included that enhance the stability of the space-time finite element method. Nonlinear discontinuity-capturing operators are presented that result in more accurate capturing of wave fronts in transient solutions while maintaining the high-order accuracy of the underlying linear algorithm in smooth regions. Stability of the discontinuity-capturing formulation is proved and a convergence proof is given for a particular choice of the discontinuity-capturing operator. Numerical results are presented that demonstrate the effectiveness of the proposed discontinuity-capturing operators for elastodynamics.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/30084/1/0000455.pd

A unified set of single-step asymptotic annihilation algorithms for structural dynamics

When solving the equations of structural dynamics using direct time integration methods, algorithmic damping is useful in controlling spurious high-frequency oscillations. Ideally, an algorithm should possess asymptotic annihilation of the high-frequency response, i.e., spurious oscillations in the high frequencies are eliminated after one time step. In this paper, a new class of asymptotic annihilation algorithms for structural dynamics is presented that possesses little numerical dissipation in the low-frequency regime. The algorithms are based upon using finite elements in time. The displacement and velocity fields are interpolated independently using time-discontinuous functions. The equations of motion, displacement-velocity compatibility and the time continuity of displacement and velocity are satisfied weakly. Asymptotic annihilation is achieved by choosing the displacement and velocity fields to be of equal order. Algorithms of any desired order of temporal accuracy can be obtained by appropriate choice of the finite element interpolations in time. An analysis of the proposed class of algorithms is given proving the asymptotic annihilation property and the spectral equivalence of the algorithms to the upper diagonal of the Pade approximation table. Results from finite difference analyses are presented showing the spectral behavior of the algorithms as well as their dissipation and dispersion properties in the low frequency regime.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/31716/1/0000652.pd

Explicit momentum conserving algorithms for rigid body dynamics

Two new explicit time integration algorithms are presented for solving the equations of motion of rigid body dynamics that identically preserve angular momentum in the absence of applied torques. This is achieved by expressing the equations of motion in conservation form. Both algorithms also eliminate the need for computing the angular acceleration. The first algorithm employs a one-pass predictor-corrector scheme while the second algorithm is based upon the staggered time integration approach of Park. Numerical results are presented comparing the new algorithms to the algorithms of Simo and Wong and Park et al. The predictor-corrector algorithm is shown to suffer weak instabilities while the staggered conserving algorithm exhibits improved performance compared to the staggered algorithm of Park et al.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/29853/1/0000200.pd

Analysis and optimal design of layered composites with high stiffness and high damping

AbstractIn this paper we investigate the design of composite materials with simultaneously high stiffness and high damping. We consider layered composite materials with parallel plane layers made of a stiff constituent and a lossy polymer. We analyze the response of these composites to a dynamic load with an arbitrary direction. Using the viscoelastic correspondence principle and linear frequency domain viscoelastic models, we derive an expression for the effective complex modulus of layered composites of infinite size at infinitesimal strains. The dependence of the effective dynamic modulus and loss factor on the geometrical parameters and on the tensile and bulk loss factors of the lossy constituent is analyzed. Moreover we determine the magnitude of the strains in the lossy constituent and demonstrate that the combination of high stiffness and high damping of these composites is due to the high normal and/or shear strains in the lossy material. We use nonlinear constrained optimization to design layered composites with simultaneously high stiffness and high damping while constraining the strains in the polymer. To determine the range of validity of the linear viscoelastic model, simulations using finite deformations models are compared to the theoretical results. Finally, we compute the effective properties of composites of finite size using finite element methods and determine the minimum size required to approach the formulae derived for composites of infinite size

A family of single-step Houbolt time integration algorithms for structural dynamics

A new family of implicit, single-step time integration methods is presented for solving structural dynamics problems. The proposed method is unconditionally stable, second-order accurate and asymptotically annihilating. It is spectrally equivalent to Houbolt's method but is cast in single-step form rather than multi-step form; thus the new algorithm computationally is more convnient. An explicit predictor-corrector algorithm is presented based upon new implicit scheme. The explicit algorithm is spectrally equivalent to the central difference method. The two new algorithms are merged into an implicit-explicit method, resulting in an improved algorithm for solving structural dynamics problems composed of `soft' and `stiff' domains. Numerical results are presented demonstrating the improved performance of the new implicit-explicit method compared to previously developed implicit-explicit schemes for structural dynamics.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/31334/1/0000244.pd

A Gluing Algorithm for Network-Distributed Multibody Dynamics Simulation

The improved performance and capacity of networks has made thecombined processing power of workstation clusters a potentiallypromising avenue for solving computationally intensive problems acrosssuch distributed environments. Moreover, networks provide an idealplatform to employ heterogeneous hardware and software to solvemultibody dynamics problems. One fundamental difficulty with distributedsimulation is the requirement to couple and synchronize the distributedsimulations. This paper focuses on the algorithms necessary to coupletogether separately developed multibody dynamics modules so that theycan perform integrated system simulation. To identify a useful couplingstrategy, candidate numerical algorithms in the literature are reviewedbriefly – namely, stiff time integration, local parameterization,waveform relaxation, stabilized constraint and perturbation. Anunobtrusive algorithm that may well serve this `gluing' role ispresented. Results from numerical experiments are presented and theperformance of the gluing algorithm is investigated.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/43264/1/11044_2004_Article_352676.pd

Accurate determination of surface normal stress in viscous flow from a consistent boundary flux method

Accurate determination of surface normal stresses from numerical modeling of mantle convection is crucial in determining surface topography, geoid and gravity anomalies. With the finite element method, we have developed a consistent boundary flux (CBF) method for computing the surface stress by solving the momentum equation directly. The method has a much higher accuracy for determining surface stresses than the standard pressure smoothing method, and for typical convection problems, the CBF is about one order of magnitude more accurate than pressure smoothing. The CBF can be easily applied to a variety of types of elements and to compute a range of physical quantities including heat flow on boundaries. CBF, moreover, is a post-processing operation and is computationally inexpensive.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/30747/1/0000397.pd

Elastomer bushing response: experiments and finite element modeling

 Elastomer bushings are essential components in tuning suspension systems since they isolate vibration, reduce noise transmission, accommodate oscillatory motions and accept misalignment of axes. This work presents an experimental study in which bushings are subjected to radial, torsional and coupled radial-torsional modes of deformation. The experimental results show that the relationship between the forces and moments and their corresponding displacements and rotations is nonlinear and viscoelastic due to the nature of the elastomeric material. An interesting feature of the coupling response is that radial force decreases and then increases with torsion. The experimental results were used to assess bushing behavior and to determine the strength of radial-torsional coupling. The experimental results were also compared to finite element simulations of a model bushing. While finite element analysis predicted small displacements at the relaxed state reasonably well, the response to larger radial deformations and coupled deformations was not well captured.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/42461/1/31630025.pd

A Gluing Algorithm for Distributed Simulation of Multibody Systems

A new gluing algorithm is presented that can be used tocouple distributed subsystem models fordynamics simulation of mechanical systems. Using this gluingalgorithm, subsystem models can be analyzed attheir distributed locations, using their own independent solvers,and on their own platforms. The gluing algorithmdeveloped relies only on information available at the subsysteminterfaces. This not only enables efficientintegration of subsystem models, but also engenders modelsecurity by limiting model access only to the exposedinterface information. These features make the algorithm suitablefor a real and practical distributed simulationenvironment.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/43325/1/11071_2004_Article_5252593.pd