TY - JOUR
T1 - Optimal local approximation spaces for component-based static condensation procedures
AU - Smetana, Kathrin
AU - Patera, Anthony T.
N1 - Publisher Copyright:
© 2016 Society for Industrial and Applied Mathematics.
PY - 2016
Y1 - 2016
N2 - In this paper we introduce local approximation spaces for component-based static condensation (sc) procedures that are optimal in the sense of Kolmogorov. To facilitate simulations for large structures such as aircraft or ships, it is crucial to decrease the number of degrees of freedom on the interfaces, or "ports," in order to reduce the size of the statically condensed system. To derive optimal port spaces we consider a (compact) transfer operator that acts on the space of harmonic extensions on a two-component system and maps the traces on the ports that lie on the boundary of these components to the trace of the shared port. Solving the eigenproblem for the composition of the transfer operator and its adjoint yields the optimal space. For a related work in the context of the generalized finite element method, we refer the reader to [I. Babuška and R. Lipton, Multiscale Model. Simul., 9 (2011), pp. 373-406]. We further introduce a spectral greedy algorithm to generalize the procedure to the parameter-dependent setting and to construct a quasi-optimal parameter-independent port space. Moreover, it is shown that, given a certain tolerance and an upper bound for the ports in the system, the spectral greedy constructs a port space that yields an sc approximation error on a system of arbitrary configuration which is smaller than this tolerance for all parameters in a rich train set. We present our approach for isotropic linear elasticity, although the idea may be readily applied to any linear coercive problem. Numerical experiments demonstrate the very rapid and exponential convergence both of the eigenvalues and of the sc approximation based on spectral modes for nonseparable and irregular geometries such as an I-beam with an internal crack.
AB - In this paper we introduce local approximation spaces for component-based static condensation (sc) procedures that are optimal in the sense of Kolmogorov. To facilitate simulations for large structures such as aircraft or ships, it is crucial to decrease the number of degrees of freedom on the interfaces, or "ports," in order to reduce the size of the statically condensed system. To derive optimal port spaces we consider a (compact) transfer operator that acts on the space of harmonic extensions on a two-component system and maps the traces on the ports that lie on the boundary of these components to the trace of the shared port. Solving the eigenproblem for the composition of the transfer operator and its adjoint yields the optimal space. For a related work in the context of the generalized finite element method, we refer the reader to [I. Babuška and R. Lipton, Multiscale Model. Simul., 9 (2011), pp. 373-406]. We further introduce a spectral greedy algorithm to generalize the procedure to the parameter-dependent setting and to construct a quasi-optimal parameter-independent port space. Moreover, it is shown that, given a certain tolerance and an upper bound for the ports in the system, the spectral greedy constructs a port space that yields an sc approximation error on a system of arbitrary configuration which is smaller than this tolerance for all parameters in a rich train set. We present our approach for isotropic linear elasticity, although the idea may be readily applied to any linear coercive problem. Numerical experiments demonstrate the very rapid and exponential convergence both of the eigenvalues and of the sc approximation based on spectral modes for nonseparable and irregular geometries such as an I-beam with an internal crack.
KW - (Component-based) static condensation
KW - A priori error estimate
KW - Component mode synthesis
KW - Domain decomposition methods
KW - Finite element method
KW - Kolmogorov n-width
KW - Model reduction
KW - Reduced basis methods
UR - http://www.scopus.com/inward/record.url?scp=84994064326&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84994064326&partnerID=8YFLogxK
U2 - 10.1137/15M1009603
DO - 10.1137/15M1009603
M3 - Article
AN - SCOPUS:84994064326
SN - 1064-8275
VL - 38
SP - A3318-A3356
JO - SIAM Journal on Scientific Computing
JF - SIAM Journal on Scientific Computing
IS - 5
ER -