Optimal Local Approximation Spaces for Component-Based Static Condensation Procedures

Abstract

Abstract. 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. 1. Introduction. In the last decades numerical simulations based on partial differential equations (PDEs) have significantly gained importance in engineering applications. However, both the geometric complexity of the considered structures, such as ships, aircraft, and turbines, and the intricacy of the simulated physical phenomena often make a straightforward application of, say, the finite element (FE) method prohibitive. This is particularly true if multiple simulation requests or a real-time simulation response is desired, as in engineering design and optimization.One way to tackle such complex problems is to exploit the natural decomposition of the structures into components and apply static condensation (sc) to obtain a (Schur complement) system of the size of the degrees of freedom (DOFs) on all interfaces or ports in the system. To mitigate the computational costs for the required PDE solvers in the interior of the component, model order reduction procedures may be applied. One popular approach is component mode synthesis (CMS), introduced in [4,21], which uses an approximation based on the eigenmodes of local constrained eigenvalue problems. The static condensation reduced basis element (scRBE) method [22,23]

Static Condensation Optimal Port/Interface Reduction and Error Estimation for Structural Health Monitoring

“…Recently, port spaces that are optimal in the sense of Kolmogorov and thus minimize the approximation error among all port spaces of the same dimension have been introduced in [24]. The approach in [24] generalizes the idea of separation of variables by connecting two components at the port for which we wish to construct the port space and consider the space of all local solutions of the partial differential equation (PDE) with arbitrary Dirichlet boundary conditions on the ports that lie on the boundary of the two-component system. From separation of variables we anticipate an exponential decay (of the higher modes) of the Dirichlet boundary conditions to the interior of the system.…”

“…In [6] it has been shown that by employing methods from randomized numerical linear algebra an extremely accurate approximation of those optimal port spaces can be computed in close to optimal computational complexity. To account for variations in a material or geometric parameter in [24] a parameter-independent port space is generated from the optimal parameter-dependent port spaces via a spectral greedy algorithm. It is further numerically demonstrated in [24] that the optimal port spaces often outperform other approaches such as Legendre polynomials [13] or empirical modes [7]; also an exponential convergence can be observed.…”

“…To account for variations in a material or geometric parameter in [24] a parameter-independent port space is generated from the optimal parameter-dependent port spaces via a spectral greedy algorithm. It is further numerically demonstrated in [24] that the optimal port spaces often outperform other approaches such as Legendre polynomials [13] or empirical modes [7]; also an exponential convergence can be observed. Finally, those optimal port modes have been used in structural integrity management of offshore structures in [17] and optimal local approximation have been exploited in the context of data assimilation in [26].…”

“…In this article we want to investigate the applicability of optimal port spaces for structural health monitoring and more specifically extend the concept of [24] to geometry changes. First, we show how to construct one port space for several different geometries such as a beam and a beam with a crack or hole via a spectral greedy algorithm.…”

“…In section 2 we introduce the problem setting and recall the algebraic (port reduced) static condensation procedure. Subsequently, we recall the optimal port spaces introduced in [24] in section 3. In section 4 we propose quasi-optimal port spaces for parametrized problems including geometry changes such as from a beam to a beam with a crack.…”