Algorithms by Design: Part I—On the Hidden Point Collocation Within LMS Methods and Implications for Nonlinear Dynamics Applications

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

doi:10.1080/15502280802365873

Cited by 19 publications

(16 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In particular, in the sense of applications to computational dynamics, are the algorithm numerical attributes such as structure preservation and/or energy-momentum type conservation properties inherited within the parent numerically non-dissipative algorithm design, and are the controllable numerically dissipative algorithm designs the direct extensions of these underlying parent numerically non-dissipative algorithm designs? Other very important and subtle issues as related to time level consistency of an algorithm (for example, every time integration algorithm satisfies the discrete equation of motion only at a particular time point, and this time level aspect is not well known for the general class ofLMS methods to-date [59]), evaluation of the nonlinear operator with parameters inside or outside the operator, the adverse effect of accelerations from the previous time step upon the computation of the unknowns in the current time step, etc., play a critical role and all relevant factors should be very clearly understood. All these are described in the next chapter in a unified setting dealing with a generalized single step single solve (GSSSS) framework of algorithms and designs.…”

Section: Guidelines and Typical Criteria For The Selection Of A Time mentioning

confidence: 99%

“…Of noteworthy importance are also issues related to time level consistency of algorithms within the GSSSS unified framework. That is, for every time integration algorithm a collocation time point exists which satisfies the discrete equation of motion; and it is essential that the various terms in the semi-discrete equation of motion be carefully evaluated at this particular time level so that it provides the necessary and theoretically expected secondorder time accuracy for all the algorithm unknowns [59]. Failure to recognize these issues will cause misinterpretation and/or improper implementation leading to poor analysis results and conclusions.…”

Section: Time Discretization and The Total Energy Framework: Linear Dmentioning

confidence: 99%

“…whereq(t n+W 1 ) =ã =q n+W 1 6 and q(t n+W 1 ) =q = q n+W 3 + t (W 1 − W 3 )q n ; see Hoitink et al [59] for discussions about the time level consistency of the discrete equation of motion. By taking the summation for A = 1, 2, .…”

Section: Discrete Total Linear Momentum Conservationmentioning

confidence: 99%

See 2 more Smart Citations

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

View full text Add to dashboard Cite

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...

show abstract

Section: Guidelines and Typical Criteria For The Selection Of A Time mentioning

confidence: 99%

Section: Time Discretization and The Total Energy Framework: Linear Dmentioning

confidence: 99%

See 1 more Smart Citation

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

View full text Add to dashboard Cite

show abstract

“…Consider the following non-linear ODE in time [21] of time differential operators, the methods of approximation based on time integral forms are considered: those…”

Section: Model Problemmentioning

confidence: 99%

Methods of Approximation in hpk Framework for ODEs in Time Resulting from Decoupling of Space and Time in IVPs

Surana¹,

Euler²,

Reddy³

et al. 2011

AJCM

View full text Add to dashboard Cite

The present study considers mathematical classification of the time differential operators and then applies methods of approximation in time such as Galerkin method ( GM ), Galerkin method with weak form ( / GM WF ), Petrov-Galerkin method ( PGM ), weighted residual method (WRY ), and least squares method or process ( LSM or LSP ) to construct finite element approximations in time. A correspondence is established between these integral forms and the elements of the calculus of variations: 1) to determine which methods of approximation yield unconditionally stable (variationally consistent integral forms, VC ) computational processes for which types of operators and, 2) to establish which integral forms do not yield unconditionally stable computations (variationally inconsistent integral forms, VIC ). It is shown that variationally consistent time integral forms in hpk framework yield computational processes for ODEs in time that are unconditionally stable, provide a mechanism of higher order global differentiability approximations as well as higher degree local approximations in time, provide control over approximation error when used as a time marching process and can indeed yield time accurate solutions of the evolution. Numerical studies are presented using standard model problems from the literature and the results are compared with Wilson's  method as well as Newmark method to demonstrate highly meritorious features of the proposed methodology.

show abstract

“…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

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Algorithms by Design: Part I—On the Hidden Point Collocation Within LMS Methods and Implications for Nonlinear Dynamics Applications

Cited by 19 publications

References 12 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

Methods of Approximation in hpk Framework for ODEs in Time Resulting from Decoupling of Space and Time in IVPs

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

Contact Info

Product

Resources

About