Space–time elements for the shock wave propagation problem

Cella, A.; Lucchesi, M.; Pasquinelli, Giuseppe

doi:10.1002/nme.1620151005

Cited by 19 publications

(5 citation statements)

References 7 publications

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“…The concept of space-time finite elements was first proposed by Fried [11] and Oden [12] in 1969. Since then the space-time marching procedure has been applied to problems in solid mechanics [13][14][15][16] and heat conduction. Varoglu and Finn were the first to apply the spacetime finite element procedure to the convection-diffusion [17,18] and Burgers equations [19].…”

Section: Discussionmentioning

confidence: 99%

“…Let be a vector of dependent variables ( , u, T in strong form and , u, T, x x , q x in the weak form) and let h ∈ V be an approximation of in n xt in which V is the appropriate approximation space. Let (12) represent either (5)- (7) or (8)-(11), then we have, (13) where E is residual or error function vector due to the approximation h . Based on Reference [1], a STVC integral form of (13) can be constructed using space-time least squares finite element process (STLSFEP) and we have the following.…”

Section: Space-time Least Squares Formulation (Stlsf)mentioning

confidence: 99%

See 1 more Smart Citation

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: Discussionmentioning

confidence: 99%

Section: Space-time Least Squares Formulation (Stlsf)mentioning

confidence: 99%

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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show abstract

“…With the introduction of discontinuous temporal ®nite elements [13±15], novel integration schemes which lead to stable and accurate results can be derived. An interesting application to a non-linear problem was demonstrated by Cella et al [16]. Although numerical properties of the ensuing algorithm for handling nonlinear analysis are problem-dependent, generally, there are no guarantees of uniqueness and stability of the solutions, this procedure did show success in a number of cases.…”

Section: Introductionmentioning

confidence: 97%

A Space–time Finite Element Method for Elasto-Plastic Shock Dynamics

Han

1999

Journal of Sound and Vibration

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“…[23], were developed and applied to engineering problems with an acceptable numerical cost. Some authors proposed some applications of space-time finite elements for elastodynamics such as linear viscoelasticity in the case of a 1D problem [52], a nonlinear 1D problem [10] or in the case of a linear 1D beam and truss or 2D plane strain problem [4,5]. Other applications than elastodynamics were also proposed, such as free surface problems (see [7]), compressible fluid flows [31], heat transfer problems [9,12], advection-diffusion equations [45].…”

Section: Introductionmentioning

confidence: 99%