Modeling dimensionally-heterogeneous problems: analysis, approximation and applications

In the present work a general theoretical framework for coupled dimensionally-heterogeneous partial differential equations is developed. This is done by recasting the variational formulation in terms of coupling interface variables. In such a general setting we analyze existence and uniqueness of solutions for both the continuous problem and its finite dimensional approximation. This approach also allows the development of different iterative substructuring solution methodologies involving dimensionally-homogeneous subproblems. Numerical experiments are carried out to test our theoretical results.

[1]  Alfio Quarteroni,et al.  Numerical Treatment of Defective Boundary Conditions for the Navier-Stokes Equations , 2002, SIAM J. Numer. Anal..

[2]  Pablo J. Blanco,et al.  A variational approach for coupling kinematically incompatible structural models , 2008 .

[3]  P. Blanco,et al.  Sensitivity analysis in kinematically incompatible models , 2009 .

[4]  F. Migliavacca,et al.  Multiscale modelling in biofluidynamics: application to reconstructive paediatric cardiac surgery. , 2006, Journal of biomechanics.

[5]  Monique Dauge,et al.  Spectral Methods for Axisymmetric Domains , 1999 .

[6]  Fabio Nobile,et al.  Robin-Robin preconditioned Krylov methods for fluid-structure interaction problems , 2009 .

[7]  Pablo J. Blanco,et al.  Assessing the influence of heart rate in local hemodynamics through coupled 3D‐1D‐0D models , 2010 .

[8]  Alfio Quarteroni,et al.  A Domain Decomposition Method for Advection-Diffusion Processes with Application to Blood Solutes , 2002, SIAM J. Sci. Comput..

[9]  Pablo J. Blanco,et al.  Multidimensional modelling for the carotid artery blood flow , 2006 .

[10]  P. Blanco,et al.  A unified variational approach for coupling 3D-1D models and its blood flow applications , 2007 .

[11]  G E Karniadakis,et al.  LARGE‐SCALE SIMULATION OF THE HUMAN ARTERIAL TREE , 2009, Clinical and experimental pharmacology & physiology.

[12]  Gianluigi Rozza,et al.  Real-Time Reliable Simulation of Heat Transfer Phenomena , 2009 .

[13]  Pablo J. Blanco,et al.  Iterative strong coupling of dimensionally heterogeneous models , 2009 .

[14]  Andrea Toselli,et al.  Domain decomposition methods : algorithms and theory , 2005 .

[15]  A. Quarteroni,et al.  Numerical Approximation of Partial Differential Equations , 2008 .

[16]  A. Quarteroni,et al.  On the coupling of 3D and 1D Navier-Stokes equations for flow problems in compliant vessels , 2001 .

[17]  Alfio Quarteroni,et al.  Analysis of a Geometrical Multiscale Model Based on the Coupling of ODE and PDE for Blood Flow Simulations , 2003, Multiscale Model. Simul..

[18]  A. Eringen,et al.  Elastodynamics, Vol. II, Linear Theory , 1978 .

[19]  Barry F. Smith,et al.  Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations , 1996 .

[20]  A. Quarteroni,et al.  Fluid–structure algorithms based on Steklov–Poincaré operators , 2006 .

[21]  I. Babuska.  The finite element method with Lagrangian multipliers , 1973 .

[22]  Barry Smith,et al.  Domain Decomposition Methods for Partial Differential Equations , 1997 .

[23]  P. Blanco,et al.  On the potentialities of 3D-1D coupled models in hemodynamics simulations. , 2009, Journal of biomechanics.

[24]  A. Quarteroni,et al.  Convergence analysis of a subdomain iterative method for the finite element approximation of the coupling of Stokes and Darcy equations , 2004 .