IUTAM Symposium on Discretization Methods for Evolving Discontinuities (eBook)

Preface; Meshless Finite Element Methods: Meshless discretisation of nonlocal damage theories, by H. Askes, T. Bennett and S. Kulasegaram; Three-dimensional non-linear fracture mechanics by enriched meshfree methods without asymptotic enrichment, by S. Bordas, C. Zi and T. Rabczuk; Accounting for weak discontinuities and moving boundaries in the context of the Natural Element Method and model reduction techniques, by F. Chinesta, F. Cueto, P. Joyot and P. Villon; Discontinuous Galerkin Methods: Modeling evolving discontinuities with spacetime discontinuous Galerkin methods, by R. Abedi, S.-H. Chung, M.A. Hawker, J. Palaniappan and R.B. Haber; Analysis of a finite element formulation for modelling phase separation G.N. Wells and K. Garikipati; Finite Element Methods with Embedded Discontinuities: Recent developments in the formulation of finite elements with embedded strong discontinuities, by F. Armero and C. Linder; Evolving material discontinuities: Numerical modeling by the Continuum Strong Discontinuity Approach (CSDA), by J. Oliver, A.E. Huespe, S. Blanco and D.L. Linero; A 3D cohesive investigation on branching for brittle materials, by R.C. Yu, A. Pandolfi and M. Ortiz; Partition-of-Unity Based Finite Element Methods: On applications of XFEM to dynamic fracture and dislocations, by T. Belytschko, J.-H. Song, H. Wang and R. Gracie; Some improvements of XFEM for cracked domains, by E. Chahine, P. Laborde, J. Pommier, Y. Renard and M. Salaün; 2D X-FEM simulation of dynamic brittle crack propagation, by A. Combescure, A. Gravouil, H. Maigre, J. Réthoré and D. Gregoire; A numerical framework to model 3-D fracture in bone tissue with application to failure of the proximal femur, by T.C. Gasser and G.A. Holzapfel; Application of X-FEM to 3D real cracks and elastic-plastic fatigue crack growth, by A. Gravouil, A. Combescure, T. Elguedj, E. Ferrié, J.-Y. Buffière and W. Ludwig; Accurate simulation offrictionless and frictional cohesive crack growth in quasi-brittle materials using XFEM B.L. Karihaloo and Q.Z. Xiao; On the application of Hansbo’s method for interface problems, by B. Kuhi, Ph. Jãger, J. Mergheim and P. Steinmann; An optimal explicit time stepping scheme for cracks modeled with X-FEM, by T. Menouillard, N. Moës and A. Combescure; Variational Extended Finite Element Model for cohesive cracks: Influence of integration and interface law, by G. Meschke, P. Dumstorff and W. Fleming; An evaluation of the accuracy of discontinuous finite elements in explicit dynamic calculations, by J.J. C. Remmers, R. de Borst, A. Needleman; A discrete model for the propagation of discontinuities in a fluid-saturated medium, by J. Réthoré, R. de Borst and M.-A. Abellan; Single domain quadrature techniques for discontinuous and non-linear enrichments in local Partition of Unity FEM, by G. Ventura; Other Discretization Methods: Numerical determination of crack stress and deformation fields in gradient elastic solids, by G.F. Karlis, S.V. Tsinopoulos, D. Polyzos and Th.E. Beskos; The variational formulation of brittle fracture: Numerical implementation and extensions, by B. Bourdin; Measurement and identification techniques for evolving discontinuities, by F. Hild, J. Réthoré and S. Roux; Conservation under incompatibility for fluid-solid-interaction problems: The NPCL method, by E.H. van Brummelen and R. de Borst; Author Index; Subject Index.

"Conservation under Incompatibility for Fluid-Solid-Interaction Problems: the NPCL Method (p. 414-415)

E.H. van Brummelen and R. de Borst

Summary. Finite-element discretizations of ?uid-solid-interaction problems only trivially preserve the conservation properties of the underlying problem under restrictive compatibility conditions on the approximation spaces for the ?uid and the solid. The present work introduces a new general method for enforcing interface conditions that maintains the conservation properties under incompatibility. The method is based on a nonlinear variational projection of the velocity ?eld to impose the kinematic condition, and a consistent evaluation of the load functional that accounts for the dynamic condition. Numerical results for a projection problem are presented to illustrate the properties of the method.

Key words: ?uid-solid interaction, incompatibility, conservation, space-time ?niteelement methods.

1 Introduction

The numerical solution of ?uid-solid-interaction problems has prominence in many scienti?c and engineering disciplines. The interaction is induced by interface conditions, which prescribe continuity of displacements and tractions across the ?uid-solid interface. If the approximation spaces for the ?uid and the solid in the discretization are compatible, i.e., if the ?uid and the solid have identical meshes and orders of approximation at the interface, then the enforcement of these continuity requirements is trivial. However, in many instances, it is necessary to allow for incompatible approximation spaces.

For instance, the meshes for the ?uid and solid subsystems may have been generated by di?erent analysts. Moreover, the disparate regularity properties of the ?uid and solid solutions typically prompt distinct approximation spaces. An important characteristic of ?uid-solid-interaction problems pertains to their conservation properties: on account of the continuity of tractions and displacements, mass, momentum and energy are conserved at the interface and, accordingly, the interface does not appear in the conservation statements for the aggregated system.

However, incompatibility impedes continuity of tractions and displacements across the interface in the discrete approximation. Consequently, incompatible ?nite-element discretizations of ?uid-solidinteraction problems do not generally preserve the conservation properties of the underlying continuum problem. Current coupling strategies for ?uid-solid interaction are in general nonconservative.

The change in a conserved quantities in the interior of the ?uid and solid domains via the interface can be expressed as an inner product on the interface. Conservation requires that this inner products evaluates to the same value at both sides of the interface. Most coupling methods however fail to identify the inner products. The methodology presented in [3] identi?es the inner products, but the inner products do not properly represent the change in the interior of the domains.

In this work we present a new general coupling method for ?uid-solidinteraction problems that preserves the conservation properties under incompatibility. The method comprises three complementary primitives: a suitable nonlinear variational projection to impose the kinematic condition, representation of the load functional in the velocity trace space of the ?uid, and a consistent evaluation of the load functional to account for the dynamic condition. We refer to the approach concisely as the NPCL (Nonlinear variational Projection with Consistent Loading) method."