Non‐weak/strong solutions in gas dynamics: a C11 p‐version STLSFEF in Eulerian frame of reference using ρ, u,p primitive variables

This paper presents a space–time least squares finite element formulation of one‐dimensional transient Navier–Stokes equations (governing differential equations: GDE) for compressible flow in Eulerian frame of reference using ρ, u, p as primitive variables with C¹¹ type p‐version hierarchical interpolations in space and time. Time marching procedure is utilized to compute time evolutions for all values of time. For high speed gas dynamics the C¹¹ type interpolations in space and time possess the same orders of continuity in space and time as the GDE. It is demonstrated that with this approach accurate numerical solutions of Navier–Stokes equations are possible without any assumptions or approximations. In the approach presented here SUPG, SUPG/DC, SUPG/DC/LS operators are neither used nor needed. Time accurate numerical simulations show resolution of shock structure (i.e. shock speed, shock relations and shock width) to be in excellent agreement with the analytical solutions. The role of diffusion i.e. viscosity (physical or artificial) and thermal conductivity on shock structure is demonstrated. Riemann shock tube is used as a model problem. True time evolutions are reported beginning with the first time step until steady shock conditions are achieved. In this approach, when the computed error functionals become zero (computationally), the computed non‐weak solutions have characteristics as those of the strong solutions of the gas dynamics equations. Copyright © 2001 John Wiley & Sons, Ltd.     

On the singularities of Green's formula and its normal derivative,with an application to surface‐wave–body interaction problems

The paper presents the non‐singular forms of Green's formula and its normal derivative of exterior problems for three‐dimensional Laplace's equation. The main advantage of these modified formulations is that they are amenable to solution by directly using quadrature formulas. Thus, the conventional boundary element approximation, which locally regularizes the singularities in each element, is not required. The weak singularities are treated by both the Gauss flux theorem and the property of the associated equipotential body. The hypersingularities are treated by further using the boundary formula for the associated interior problems. The efficacy of the modified formulations is examined by a sphere, in an infinite domain, subject to Neumann and Dirichlet conditions, respectively. The modified integral formulations are further applied to a practical problem, i.e. surface‐wave–body interactions. Using the conventional boundary integral equation formulation is known to break down at certain discrete frequencies for such a problem. Removing the ‘irregular’ frequencies is performed by linearly combining the standard integral equation with its normal derivative. Computations are presented of the added‐mass and damping coefficients and wave exciting forces on a floating hemisphere. Comparing the numerical results with that by other approaches demonstrates the effectiveness of the method. Copyright © 2000 John Wiley & Sons, Ltd.     

Order reduction in computational inelasticity: Why it happens and how to overcome it—The ODE‐case of viscoelasticity

Time integration is the numerical kernel of inelastic finite element calculations, which largely determines their accuracy and efficiency. If higher order Runge–Kutta (RK) methods, p≥3, are used for integration in a standard manner, they do not achieve full convergence order but fall back to second‐order convergence. This deficiency called order reduction is a longstanding problem in computational inelasticity. We analyze it for viscoelasticity, where the evolution equations follow ordinary differential equations. We focus on RK methods of third order. We prove that the reason for order reduction is the (standard) linear interpolation of strain to construct data at the RK‐stages within the considered time interval. We prove that quadratic interpolation of strain based on t_n, t_{n + 1} and, additionally, t_{n ? 1} data implies consistency order three for total strain, viscoelastic strain and stress. Simulations applying the novel interpolation technique are in perfect agreement with the theoretical predictions. The present methodology is advantageous, since it preserves the common, staggered structure of finite element codes for inelastic stress calculation. Furthermore, it is easy to implement, the overhead of additional history data is small and the computation time to obtain a defined accuracy is considerably reduced compared with backward Euler. Copyright © 2011 John Wiley & Sons, Ltd.