k-Version of finite element method in gas dynamics: Higher-order global differentiability numerical solutions

International Journal for Numerical Methods in Engineering

Volumn 69, Issue 6, 2007, Pages 1109-1157

  Surana, K S ^a   Allu, S ^a   Tenpas, P W ^a   Reddy, J N ^b  

^a University of Kansas   (United States)

^b Department of Electrical and Computer Engineering   (United States)

Author keywords

Gas dynamics; Higher order global differentiability; k version finite element; Least squares; Riemann shock tube; Space time integral forms; Space time variational consistency

Indexed keywords

COMPRESSIBLE FLOW; DIFFERENTIAL EQUATIONS; FINITE ELEMENT METHOD; INTEGRAL EQUATIONS; MATHEMATICAL MODELS; NAVIER STOKES EQUATIONS; SHOCK TUBES; VISCOUS FLOW;

HIGHER-ORDER GLOBAL DIFFERENTIABILITY; K-VERSION FINITE ELEMENT METHOD; RIEMANN SHOCK TUBE; SPACE-TIME INTEGRAL FORMS; SPACE-TIME VARIATIONAL CONSISTENCY;

GAS DYNAMICS;

COMPRESSIBLE FLOW; DIFFERENTIAL EQUATIONS; FINITE ELEMENT METHOD; GAS DYNAMICS; INTEGRAL EQUATIONS; MATHEMATICAL MODELS; NAVIER STOKES EQUATIONS; SHOCK TUBES; VISCOUS FLOW;

EID: 33846667270     PISSN: 00295981     EISSN: 10970207     Source Type: Journal    
DOI: 10.1002/nme.1801     Document Type: Article

References (45)

1. Surana KS, Allu S, Reddy JN. The k-version of finite element method for IVP: mathematical and computational framework. International Journal for Numerical Methods in Engineering 2004, in review.
2. Lewis RW, Bruch Jr JC. An application of least squares to one-dimensional transient problems. International Journal for Numerical Methods in Engineering 1974; 8:633-647.
3. Donea J. A Taylor-Galerkin method for convective transport problems. International Journal for Numerical Methods in Engineering 1984; 20:101-119.
4. Lhner R, Morgan K, Zienkiewicz OC. The solution of non-linear hyperbolic equation systems by the finite element method. International Journal for Numerical Methods in Fluids 1984; 4:1043-1063.
5. Donea J, Giuliani S, Laval H, Quartapelle L. Time-accurate solution of advection-diffusion problems by finite elements. Computer Methods in Applied Mechanics and Engineering 1984; 45:123-145.
6. Kujawski J, Wiberg NE. Least-sqaures schemes for times integration of thermal problems. International Journal for Numerical Methods in Engineering 1987; 24:159-175.
7. Carey GF, Jiang BN. Least squares finite elements for the first-order hyperbolic systems. International Journal for Numerical Methods in Engineering 1988; 26:81-93.
8. Jiang BN, Carey GF. A stable least-squares finite element method for non-linear hyperbolic problems. International Journal for Numerical Methods in Fluids 1998; 8:933-942.
9. Sampaio PAB De. A Petrov-Galerkin/modified operator formulation for convection-diffusion problems. International Journal for Numerical Methods in Engineering 1990; 30:331-347.
10. Sampaio PAB De. A Petrov-Galerkin formulation for the incompressible Navier-Stokes equations using equal order interpolation for velocity and pressure. International Journal for Numerical Methods in Engineering 1991; 31:1135-1149.
11. Fried I. Finite-element analysis of time-dependent phenomena. American Institute of Aeronautics and Astronautics Journal 1969; 7:1170-1172.
12. Oden JT. A general theory of finite element. Parts I and II. International Journal for Numerical Methods in Engineering 1969; 1:205-221, 247-259.
13. Cella A, Lucchesi M, Pasquinelli G. Space-time elements for the shock wave propagation problem. International Journal for Numerical Methods in Engineering 1980; 15:1475-1488.
14. Hulbert GM, Hughes TJR. Space-time finite element methods for second-order hyperbolic equations. Computer Methods in Applied Mechanics and Engineering 1980; 84:327-348.
15. Burch Jr JC, Zyvoloski G. Transient two-dimensional heat conduction problems solved by the finite element method. International Journal for Numerical Methods in Engineering 1974; 8:481-494.
16. Yu JR, Hsu TR. Analysis of heat conduction in solids by space-time finite element method. International Journal for Numerical Methods in Engineering 1985; 21:2001-2012.
17. Varoglu E, Finn WDL. A finite element method for diffusion-convection equation with constant coefficients. Advances Water Resources 1978; 1:337-343.
18. Varoglu E, Finn WDL. Finite elements incorporating characteristics for one-dimensional diffusion-convection equation. Journal of Computational Physics 1980; 34.
19. Varoglu E, Finn WDL. Space-time finite elements incorporating characteristics for the Burgers' equation. International Journal for Numerical Methods in Engineering 1980; 16:171-184.
20. Lam DCL. Comparison of finite-element and finite difference methods for nearshore advection-diffusion transport models. Finite Elements in Water Resources 1977; 1.115-1.129.
21. Richter GR. A finite element method for time-dependent convection-diffusion equations. Journal of Computational Mathematics 1990; 54:81-106.
22. Hughes TJR, Franca LP, Hulbert GM. A new finite element formulation for computational fluid dynamics: VIII. The Galerkin/least squares method for advection-diffusive equations. Computer Methods in Applied Mechanics and Engineering 1989; 73:173-189.
23. Shakib F, Hughes TJR. A new finite element formulation for computational fluid dynamics: IX. Fourier analysis of space-time Galerkin/least squares algorithms. Computer Methods in Applied Mechanics and Engineering 1991; 87:35-58.
24. Nguyen H, Reynen J. A space-time least-square finite element scheme for advection-diffusion equations. Computer Methods in Applied Mechanics and Engineering 1984; 42:331-342.
25. Nguyen H, Reynen J. A Space-Time Finite Element Approach to Burgers' Equation in Numerical Methods for Non-linear Problems, vol. 2. Pineridge Press: Swansea, 1984; 718-728.
26. Kececioglu I, Rubinsky B. A mixed-variable continuously deforming finite element method for parabolic evolution problems. Part I: The variational formulation for single evolution equation. International Journal for Numerical Methods in Engineering 1989; 28:2583-2607.
27. Kececioglu I, Rubinsky B. A mixed-variable continuity deforming finite element method for parabolic evolution problems. Part II: The coupled problem of phase change in porous media. International Journal for Numerical Methods in Engineering 1989; 28:2609-2634.
28. Kececioglu I, Rubinsky B. A mixed-variable continuously deforming finite element method for parabolic evolution problems. Part III: Numerical implementation and computational results. International Journal for Numerical Methods in Engineering 1989; 28:2715-2760.
29. Bell BC, Surana KS. p-Version least squares finite element formulation for two dimensional isothermal and non-isothermal incompressible non-Newtonian flow. International Journal for Numerical Methods in Engineering 1994; 18:127-162.
30. Bell BC, Surana KS. A space-time coupled p-version least squares finite element formulation for unsteady fluid dynamics problems. International Journal for Numerical Methods in Engineering 1994; 37:3545-3569.
31. Surana KS, Bona M. Non-weak/strong solutions of linear and non-linear hyperbolic and parabolic equations resulting from a single conservative law. International Journal of Computational Engineering Science 2000; 1(2):299-330.
32. 11 p-version STLSFEF in Lagrangian frame of reference using ρ, u, P primitive variables. International Journal for Numerical Methods in Engineering 2002; 53:1025-1050.
33. 11 p-version STLSFEF in Eulerian frame of reference using ρ, u, P primitive variables. International Journal for Numerical Methods in Engineering 2002; 53:1051-1099.
34. 11 p-version STLSFEF in Lagrangian frame of reference using ρ, u, T primitive variables. International Journal of Computational Engineering Science 2001; 2(3):357-382.
35. 11 p-version STLSFEF in Eulerian frame of reference using ρ, u, T primitive variables. International Journal of Computational Engineering Science 2001; 2(3):383-423.
36. Nayak HV. Solutions of classes C00 and C11 for two dimensional Newtonian and polymer flows. Ph.D. Dissertation, University of Kansas, 2001.
37. Gelfand IM, Fomin SV. Calculus of Variations. Dover: New York, 2000.
38. Mikhlin SG. Variational Methods in Mathematical Physics. Pergamon Press: New York, 1964.
39. Surana KS, Ahmadi A, Reddy JN. k-Version of finite element method for self-adjoint operators in B.V.P. International Journal of Computational Engineering Science 2002; 3(2):155-218.
40. Surana KS, Ahmadi A, Reddy JN. k-Version of finite element method for non-self-adjoint operators in B.V.P. International Journal of Computational Engineering Science 2003; 4(4):737-812.
41. Surana KS, Ahmadi A, Reddy JN. k-Version of finite element method for non-linear operators in B.V.P. International Journal of Computational Engineering Science 2004; 5(1):133-207.
42. Surana KS, Reddy JN. Strong and weak form of GDEs and choices of dependent variables in LS and STLS processes, in preparation.
43. Van Dyne DG. Nonweak/strong solutions in gasdynamics. Ph.D. Thesis, Mechanical Engineering Department, University of Kansas, 1999.
44. Jiang BN. The Least Squares Finite Element Method. Springer: New York, 1998.
45. Surana KS, Petti SR, Ahmadi AR, Reddy JN. On p-version hierarchical interpolation functions for higher-order continuity finite element models. International Journal of Computational Engineering Science 2001; 2(4):653-673.