Space-time-finite elements for two-phase models based on the theory of porous media
Final Report Abstract
The subject of this project is to develop an efficient space-time finite element scheme for the numerical modeling of porous materials. The conventional approaches of solving such problems are known as the Method of Lines (MOLs), in which a conforming finite element discretization is first applied in the spatial domain to produce a system of Ordinary Differential Equations (ODE) in time, which can be in turn solved by a certain finite difference method in time, e. g. the Euler scheme, Newmark scheme, etc. The drawback of such approaches is that they usually suffer from strong numerical dissipation and dispersion. In the case of steep gradient fields in the computational domain, unphysical oscillations in the numerical solution are usually observed. In this project, we developed a coupled space-time finite element scheme for the solution strategy of coupled problems. The finite element discretization was applied in the spatial and the temporal domain simultaneously, leading to a space-time coupled formulation. According to the type of ansatz functions employed in the numerical scheme, we developed the Time- Discontinuous Galerkin (DGT) method, in which the spatial approximations are continuous while discontinuous ones are applied in time; and the discontinuous space-time Galerkin (DGST) method, in which neither the spatial nor the temporal domain possess strong continuities. In the case of different quantities (jumps) on the element interfaces, various flux treatments are applied to weakly enforce the continuity condition. No artificial penalty term is involved in the developed numerical formulation. In addition, taken into account that the DGT approach is only suitable for solving the first-order time-dependent problems, an Embedded Velocity Integration (EVI) scheme is proposed to implicitly include the kinematic relation into the numerical integration scheme, so that the direct solution of the second-order term is circumvented to the solution of its rate term. In contrast to the conventional order-reduction technique, so that an extra equation is introduced into the governing set of equations, the resulting equation system of the EVI approach possesses the same dimension of the original equation system, which makes the new formulation very attractive. Furthermore, based on the existing EVI formulation, a generalized EVI technique equipped with a stabilization factor is proposed. By a proper choice of , the overall stability of the numerical solution can be enforced accordingly. The space-time Galerkin formulations developed in this project provide a general foundation for modeling dynamic effects in various applications. We investigated elastic waves propagating in a porous material consisting of incompressible solid phase and a barotropic fluid phase (hybrid model). In comparison with the conventional approaches, the newly proposed method has the advantage in low numerical dissipation and less numerical dispersion. The proposed approach can be applied to model realistic geological phenomena, e. g. propagation of earthquake waves, analysis of seismological waves in the oil recovery industry, etc. Dynamics at finite deformation for the hybrid model were also investigated. Large deformations in porous materials are commonly observed in biological engineering, e. g. in soft tissue, or in energy absorbing materials in the industry, e. g. cushion and crash worthiness in the automotive industry. Generally speaking, the methods developed in this project exhibit an efficient and robust scheme for the treatment of dynamic effects in multiphase materials.
Publications
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Chen, Z. & Steeb, H. [2005], A DGT-method for the numerical analysis of wave propagation in fully-saturated porous media, in V. de Gennaro, J.-M. Pereira & P. Delage, eds., Proceedings of W(H)YDOC '05, 2nd International Workshop of Young Doctors in Geomechanics, Paris, 2005 , 53-56, Ecole Nationale des Ponts et Chaussées, Paris, France.
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Chen, Z., Steeb, H. & Diebels, S. [2005], A Discontinuous Galerkin method for the dynamic analysis of fully-saturated porous media, in D. R. J. Owen, E. O. nate & B. Suárez, eds., Proceedings of COMPLASS 05 , 101-104, CIMNE, Barcelona, Spain.
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Chen, Z., Steeb, H. & Diebels, S. [2005], Space-time DG method for porous media, Proc. Appl. Math. Mech., 5, 385-386.
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Chen, Z., Steeb, H. & Diebels, S. [2006], A Time-Discontinuous Galerkin method for the dynamical analysis of porous media, Int. J. Numer. Anal. Meth. Geomech., 30, 1113-1134.
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Chen, Z., Steeb, H. & Diebels, S. [2006], Analysis of wave propagation in the fluid-saturated porous media, Proc. Appl. Math. Mech., 6, 429-430.
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Chen, Z., Steeb, H. & Diebels, S. [2007], Analysis of wave propagation in the fluid-saturated porous media, Proc. Appl. Math. Mech., 6, 429-430.
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Chen, Z., Steeb, H. & Diebels, S. [2007], Dynamic analysis of porous materials: Numerical simulation with adaptive space-time FEM, Proc. Appl. Math. Mech., submitted for publication.
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Chen, Z., Steeb, H. & Diebels, S. [2007], Space-time Galerkin method for coupled problems in porous media, in Proceedings of coupled problems, 291-295, CIMNE, Barcelona, Spain.
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Chen, Z., Steeb, H. & Diebels, S. [2007], Waves in fully saturated porous media investivated by adaptive space-time finite element method, in K. Runesson & P. D`iez, eds., Adaptive modeling and simulation, Proceedings of III International conference on adaptive modeling and simulation, 69-72, CIMNE, Barcelona, Spain.
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Chen, Z., Steeb, H. & Diebels, S. [2008], A new hybrid velocity integration method applied to elastic wave propagation, Int. J. Numer. Meth. Eng., 74(1), 56-79.
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Chen, Z. [2008], Coupled space-time discontinuous Galerkin methods for dynamical modeling in porous media, Dissertation, Chair of Applied Mechanics, Saarland University.
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Steeb, H. [2008], Non-equilibrium processes in porous media, Habilitation, submitted to the Faculty of Chemistry, Pharmacy, Bio-and Materials Science, Saarland Universtiy.
