The k-Version of Finite Element Method for Initial Value Problems: Mathematical and Computational Framework

Surana, Karan S.; Reddy, J. N.; Allu, Srikanth

doi:10.1080/15502280701252321

Cited by 25 publications

(51 citation statements)

References 41 publications

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“…In view of References [1,29,30] and Surana et al [31][32][33][34][35][36], we only consider space-time coupled processes. Space-time decoupled methods are contrary to the physics and mathematics of IVP and do not permit development of a rigorous mathematical framework [1], and hence are not entertained here.…”

Section: Objectivementioning

confidence: 99%

“…In a recent paper Surana et al [1] presented a general mathematical and computational framework for all IVPs regardless of their origin or field of application. In this section, we summarize the main features of this framework before presenting its application to gas dynamics.…”

Section: General Strategy and Considerationsmentioning

confidence: 99%

“…Thus, now we have a four-dimensional space x, y, z and t. The result is that (1) for one-dimensional IVP, we have x and t, resulting in space-time strip¯ Figure 1, (2) for twodimensional IVP, we have x, y and t, resulting in space-time slab¯ Figure 2, where¯ x is the two-dimensional x, y domain (3) 4. It was shown in Reference [1] that in order to pursue space-time finite element processes for IVP, one must construct a space-time integral form using the space-time differential operator over¯ n xt which is possible based on fundamental lemma [1,[37][38][39] and then determine whether this space-time integral form is space-time variationally consistent (STVC). It has been shown that space-time variationally inconsistent (STVIC) integral forms may lead to degenerate computational processes that may produce spurious solutions for some choices of computational and physical parameters, and hence must be avoided at all cost.…”

Section: General Strategy and Considerationsmentioning

confidence: 99%

“…Concepts discussed thus far are intrinsic in all IVP regardless of their origin and fields of applications, and hence are applicable to all areas of continuum mechanics. In this paper, we first briefly discuss a general mathematical and computational framework presented by Surana et al [1] applicable to all IVP and then present high-speed compressible gas dynamics as an application.…”

Section: Introductionmentioning

confidence: 99%

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k‐version of finite element method in gas dynamics: higher‐order global differentiability numerical solutions

Surana

Allu

Tenpas

et al. 2006

Numerical Meth Engineering

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SUMMARYIn this paper, we consider and examine alternate finite element computational strategies for time-dependent Navier-Stokes equations describing high-speed compressible flows with shocks in a viscous and conducting medium, with the ultimate objective of establishing the desired features of a general mathematical and computational framework for such initial value problems (IVP) in which: (a) the numerically computed solutions are in agreement with the physics of evolution described by the governing differential equations (GDEs) i.e. the IVP, (b) the solutions are admissible in the non-discretized form of the GDEs in the pointwise sense (i.e. anywhere and everywhere) in the entire space-time domain, and hence in the integrated sense as well, (c) the numerical approximations progressively approach the same global differentiability in space and time as the theoretical solutions, (d) it is possible to time march the solutions (this is essential for efficiency as well as ensuring desired accuracy of the computed solution for the current increment of time, i.e. to minimize the error build up in the time marching process), (e) the computational process is unconditionally stable and non-degenerate regardless of the choice of discretization, nature of approximations and their global differentiability and the dimensionless parameters influencing the physics of the process, (f) there are no issues of stability, CFL number limitations and (g) the mathematical and computational methodology is independent of the nature of the space-time differential operators.We consider one-dimensional compressible flow in a viscous and conducting medium with shocks as model problems to illustrate various features of the general mathematical and computational framework used here and to demonstrate that the proposed framework is general and is applicable to all IVP. The Riemann shock tube with a single diaphragm serves as a model problem. The specific details presented in the paper discuss: (1) Choice of the form of the GDEs, i.e. strong form or weak form. medium. Extension of this work to real gas models will be presented in a separate paper. It is worth noting that when the medium is viscous and conducting, the solutions of gas dynamics equations are analytic. (5) It is also significant to note that upwinding methods based on addition of artificial diffusion such as SUPG, SUPG/DC, SUPG/DC/LS and their many variations are neither needed nor used in this present work. (6) Numerical studies are aimed at resolving the localized details of the shock structure, i.e. shock relations, shock width, shock speed, etc. as well as the over all global behaviour of the solution in the entire space-time domain. (7) Numerical studies are presented for Riemann shock tube for high Mach number flows with special emphasis also on time accuracy of the evolution which is ensured by requiring that the approximations for each increment of time satisfy non-discretized form of the GDEs in the pointwise sense, and hence in the integrated sense as well. (8) Compari...

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Section: Objectivementioning

confidence: 99%

Section: General Strategy and Considerationsmentioning

confidence: 99%

Section: General Strategy and Considerationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

k‐version of finite element method in gas dynamics: higher‐order global differentiability numerical solutions

Surana

Allu

Tenpas

et al. 2006

Numerical Meth Engineering

Self Cite

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show abstract

“…xt ) are the space-time local approximation functions [43]. With the local approximation n e h (x, t) defined by (6), the STLSP can be described as follows.…”

Section: Space-time Lsp In H P K Frameworkmentioning

confidence: 99%

Least‐squares finite element processes in h, p, k mathematical and computational framework for a non‐linear conservation law

Surana

Allu

Reddy

et al. 2008

Numerical Methods in Fluids

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SUMMARYThis paper considers numerical simulation of time-dependent non-linear partial differential equation resulting from a single non-linear conservation law in h, p, k mathematical and computational framework in which k = (k 1 , k 2 ) are the orders of the approximation spaces in space and time yielding global differentiability of orders (k 1 −1) and (k 2 −1) in space and time (hence k-version of finite element method) using space-time marching process. Time-dependent viscous Burgers equation is used as a specific model problem that has physical mechanism for viscous dissipation and its theoretical solutions are analytic. The inviscid form, on the other hand, assumes zero viscosity and as a consequence its solutions are non-analytic as well as non-unique (Russ. Math. Surv. 1962 Math. 1965; 18:697-715) in which artificial viscosity is a function of spatial discretization, which diminishes with progressively refined discretizations. The thrust of the present work is to point out that: (1) viscous form of the Burgers equation already has the essential mechanism of viscosity (which is physical), (2) with progressively increasing Reynolds (Re) number (thereby progressively reduced viscosity) the solutions approach that of the inviscid form, (3) it is possible to compute numerical solutions for any Re number (finite) within hpk framework and space-time least-squares processes, (4) the spacetime residual functional converges monotonically and that it is possible to achieve the desired accuracy, (5) mechanism for approaching the solutions of inviscid form with progressively increasing Re. Numerical studies are presented and the computed solutions are compared with published work.

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Some Remarks on the Numerical Solution of a Strain Gradient Plasticity Theory

Bayerschen

Wulfinghoff

Böhlke

2013

Proc Appl Math and Mech

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A one-dimensional strain gradient plasticity model is reviewed in the context of Finite Element Method (FEM) solvability. It is found that under certain conditions specified the resulting system of two coupled partial differential equations in space and time leads to a symmetric global stiffness matrix. This is achieved upon standard weak form approach and yields a favorable system for an efficient numerical solution.

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The k-Version of Finite Element Method for Initial Value Problems: Mathematical and Computational Framework

Cited by 25 publications

References 41 publications

k‐version of finite element method in gas dynamics: higher‐order global differentiability numerical solutions

k‐version of finite element method in gas dynamics: higher‐order global differentiability numerical solutions

Least‐squares finite element processes in h, p, k mathematical and computational framework for a non‐linear conservation law

Some Remarks on the Numerical Solution of a Strain Gradient Plasticity Theory

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