Algorithms by Design: Part II—A Novel Normalized Time Weighted Residual Methodology and Design of a Family of Symplectic-Momentum Conserving Algorithms for Nonlinear Structural Dynamics

Masuri, Siti Ujila; Hoitink, A.; Zhou, Xiangmin; Tamma, Kumar K.

doi:10.1080/15502280802575422

Cited by 14 publications

(28 citation statements)

References 20 publications

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“…We show and explain in [59,81,82], that a different representation for the update of the unkowns are preferable to help foster angular momentum conservation, thereby avoiding errors due to acceleration computations.…”

Section: Computational and Implementation Aspectsmentioning

confidence: 98%

“…The DNA parameters {W, , λ} are given as shown in Algorithms 9 and 10 for the U0 family of algorithms and the V0 family of algorithms, respectively. This approach fails to provide proper extensions for nonlinear dynamic situations in contrast to the latter two approaches that are described next [59,[80][81][82].…”

Section: Approach 1: Classical Framework-classical Time Weighted Resimentioning

confidence: 99%

“…The DNA parameters {W, , λ} are given as shown in Algorithm 9 and Algorithm 10 for the U0 family of algorithms and the V0 family of algorithms, respectively. The representations yield symplectic-momentum based algorithm designs (and, when controllable numerical dissipation is turned off, they readily yield a family of symplecticmomentum conserving algorithms in the strict sense of the single-field form of representation of the equation of motion[59,81,82]). 2.…”

mentioning

confidence: 99%

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An Overview and Recent Advances in Vector and Scalar Formalisms: Space/Time Discretizations in Computational Dynamics—A Unified Approach

Tamma

Har

Zhou

et al. 2011

Arch Computat Methods Eng

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An overview of recent advances in computational dynamics for modeling and simulation is described. The targeted objectives are towards a wide variety of science and engineering applications in particle and continuum dynamics of structures and materials which fall in this class. Starting with the supposition that in the beginning, the well known Newton's law of motion for N-body systems is given, and is a statement of the principle of balance of linear momentum, subsequently, using this as a landmark, firstly, the principal relations to various other distinctly different fundamental principles which are of primary interest here are established. Likewise, for continuum dynamics of structures, via the principle of balance of linear momentum, analogous developments are also established. Consequently, these distinctly different fundamental principles are shown to serve as the starting point for the various developments. Stemming from three distinctly different fundamental principles, we present recent advances in N-body dynamical systems, and also continuous-body dynamical systems with focus on numerical aspects in space/time discretization. The fundamental principles are the following:the Principle of Virtual Work in Dynamics, Hamilton's Principle and as an alternate (due to inconsistencies associated with Hamilton's principle), Hamilton's Law of Varying Action, and the Principle of Balance of Mechanical Energy. Both vector and scalar formalisms are described in detail with particular focus towards general numerical discretizations in space and/or time for N-body and continuumelastodynamics applications which are encountered in a wide class of holonomic-scleronomic problems. The formulations include the classical Newtonian mechanics framework with vector formalism, and new scalar formalisms with descriptive functions such as the Lagrangian, the Hamiltonian, and the Total Mechanical Energy to readily enable numerical discretizations. The concepts emanating from the present developments and distinctly different fundamental principles inherently: (1) can independently be shown to yield the strong form of the governing mathematical model equations of motion that are continuous in space and/or time together with the natural boundary conditions; the various frameworks for the case of holonomic-scleronomic systems with the mentioned limitations are indeed all equivalent, (2) can explain naturally how the weak statement of the Bubnov-Galerkin weighted-residual form that is customarily employed for discretization with vector formalism arises for both space and time, and (3) can circumvent relying upon traditional practices of conducting numerical discretizations starting either from balance of linear momentum (Newton's second law) involving Cauchy's equations of motion (governing equations) arising from continuum mechanics or via (1) and (2) above if one chooses this option. The concepts instead provide new avenues for numerical space/time discretization for continuum-dynamical systems or time discretization for N-body system...

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Section: Computational and Implementation Aspectsmentioning

confidence: 98%

Section: Approach 1: Classical Framework-classical Time Weighted Resimentioning

confidence: 99%

mentioning

confidence: 99%

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An Overview and Recent Advances in Vector and Scalar Formalisms: Space/Time Discretizations in Computational Dynamics—A Unified Approach

Tamma

Har

Zhou

et al. 2011

Arch Computat Methods Eng

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“…The last equation in (113) may be regarded as an unconstrained formulation with two dependent field variables in the time domain; thus we have the hybrid two-field problem within the Hamiltonian framework. In general, the time weighted-residual method [Tamma et al 2000;Masuri et al 2009a;2009b] or the time finite element procedure [Simkins 1981;Aharoni and Bar-Yoseph 1992] can be applied to (113) for developing time stepping schemes. In the presence of nonconservative forces such as contact forces on the contact boundary surfaces of nonholonomic systems, we have…”

Section: Space-discrete Finite Element Formulation In the Three Formamentioning

confidence: 99%

Finite element formulations via the theorem of expended power in the Lagrangian, Hamiltonian and total energy frameworks

Har

Tamma

2009

J. Mech. Mater. Struct.

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Traditionally, variational principles and variational methods have been employed in describing finite element formulations for elastodynamics applications. Here we present alternative avenues emanating from the theorem of expended power, using the differential calculus directly.We focus on scalar representations under three distinct frameworks: Lagrangian mechanics, Hamiltonian mechanics, and a new framework involving a built-in measurable quantity, called the total energy in the configuration space. All three frameworks are derivable from each other, since they represent the same physics as Newton's second law; however, the total energy framework which we advocate inherits features that are comparable and competitive to the usual Newtonian based finite element formulations, with several added advantages ideally suited for conducting numerical discretization.The present approach to numerical space-time discretization in continuum elastodynamics provides physical insight via the theorem of expended power and the differential calculus involving the distinct scalar functions: the Lagrangian ᏸ(q,q) : T Q → ‫,ޒ‬ the Hamiltonian Ᏼ( p, q) : T * Q → ‫,ޒ‬ and the total energy Ᏹ(q,q) : T Q → ‫.ޒ‬ We show that in itself the theorem of expended power naturally embodies the weak form in space, and after integrating over a given time interval yields the weighted residual form in time. Hence, directly emanating from the theorem of expended power, this approach yields three differential operators: a discrete Lagrangian differential operator, a Hamiltonian differential operator, and a total energy differential operator.The semidiscrete ordinary differential equations in time derived with our approach can be readily shown to preserve the same physical attributes as the corresponding continuous systems. This contrasts with traditional approaches, where such proofs are nontrivial or are not readily tractable.The modeling of complicated structural dynamical systems such as a rotating bar and the Timoshenko beam are shown for illustration.

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“…Finally, it is also the objective in this exposition to investigate and outline all possible outcomes of algorithm designs with energy-momentum conserving features within the GSSSS framework, which encompasses this particular class of LMS methods. Other efforts dealing with symplectic-momentum conserving attributes and related algorithm designs within this class of LMS methods are described elsewhere in [3][4][5].…”

Section: Introductionmentioning

confidence: 99%

Algorithms by design: A new normalized time‐weighted residual methodology and design of a family of energy–momentum conserving algorithms for non‐linear structural dynamics

Masuri

Hoitink

Zhou

et al. 2009

Numerical Meth Engineering

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However, traditional practices via classical time-weighted residual approaches fail to preserve the underlying physics and stability and do not serve the purposes of extensions to non-linear dynamic situations. In contrast, a new normalized time-weighted residual approach is proposed that naturally enables such extensions leading to the design of a family of conserving time operators for unconstrained conservative dynamic systems without resorting to enforcing energy constraints as in past practices. Consequently, the algorithmic stability (in the sense of energy stability) via this approach for non-linear dynamic problems is also preserved. For linear dynamic situations, it reverts to the classical paradigm. Only simple numerical illustrations are purposely presented to demonstrate the basic concepts.

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Algorithms by Design: Part II—A Novel Normalized Time Weighted Residual Methodology and Design of a Family of Symplectic-Momentum Conserving Algorithms for Nonlinear Structural Dynamics

Cited by 14 publications

References 20 publications

An Overview and Recent Advances in Vector and Scalar Formalisms: Space/Time Discretizations in Computational Dynamics—A Unified Approach

An Overview and Recent Advances in Vector and Scalar Formalisms: Space/Time Discretizations in Computational Dynamics—A Unified Approach

Finite element formulations via the theorem of expended power in the Lagrangian, Hamiltonian and total energy frameworks

Algorithms by design: A new normalized time‐weighted residual methodology and design of a family of energy–momentum conserving algorithms for non‐linear structural dynamics

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