A port-reduced static condensation reduced basis element method for large component-synthesized structures: approximation and A Posteriori error estimation
- Jens L Eftang1, 2Email author and
- Anthony T Patera1
https://doi.org/10.1186/2213-7467-1-3
© Eftang and Patera; licensee Springer. 2013
Received: 9 August 2013
Accepted: 20 December 2013
Published: 29 January 2014
Abstract
Background
We consider a static condensation reduced basis element framework for efficient approximation of parameter-dependent linear elliptic partial differential equations in large three-dimensional component-based domains. The approach features an offline computational stage in which a library of interoperable parametrized components is prepared; and an online computational stage in which these component archetypes may be instantiated and connected through predefined ports to form a global synthesized system. Thanks to the component-interior reduced basis approximations, the online computation time is often relatively small compared to a classical finite element calculation.
Methods
In addition to reduced basis approximation in the component interiors, we employ in this paper port reduction with empirical port modes to reduce the number of degrees of freedom on the ports and thus the size of the Schur complement system. The framework is equipped with efficiently computable a posteriori error estimators that provide asymptotically rigorous bounds on the error in the approximation with respect to the underlying finite element discretization. We extend our earlier approach for two-dimensional scalar problems to the more demanding three-dimensional vector-field case.
Results and Conclusions
This paper focuses on linear elasticity analysis for large structures with tens of millions of finite element degrees of freedom. Through our procedure we effectively reduce the number of degrees of freedom to a few thousand, and we demonstrate through extensive numerical results for a microtruss structure that our approach provides an accurate, rapid, and a posteriori verifiable approximation for relevant large-scale engineering problems.
Keywords
Background
For several decades the finite element (FE) method has been a popular and important tool in engineering design and analysis of systems modelled by partial differential equations (PDEs). In particular, in fields such as structural analysis and strength assessment, the FE method is in widespread use in industry through a variety of commercial software packages. Many of the structures that are subject to industrial FE analysis are composed of a large number of components — consider for example a truss bridge, a space satellite [1], or a building or vehicle frame. Such large and at first sight complicated structures pose challenges both in terms of initial manual labor related to domain modelling and meshing, and in terms of subsequent computational cost.
Component-based structures which contain many identical or similar components are often analyzed through substructuring or superelement techniques [2], which mitigate some of these issues. Mathematically, superelement techniques are based on static condensation of all FE degrees of freedom that are interior to components, and hence the size of the global but condensed linear-algebraic (Schur complement) system is equal to the number of degrees of freedom associated with component interfaces, henceforth in this paper referred to as ports. The static condensation step necessitates a large number of component-interior FE “bubble” solves — one FE solve for each degree of freedom on each port of each component — and is for this reason rather expensive; however this step is embarrassingly parallel, and is furthermore required only once for each unique component instantiation.
Model order reduction techniques can be applied to substructuring or superelement procedures in order to further reduce the computational cost. A well-known approach is the classical component mode synthesis (CMS) [3, 4], which replaces the original FE spaces for the component-interior bubble solves with spaces spanned by a few component-interior eigenmodes. As a result, the cost associated with each bubble calculation is reduced, and the formation of the global Schur complement system is consequently much less expensive.
A more recent approach, which is relevant in the context of parameter-dependent PDEs and which we for this reason consider here in this paper, is the static condensation reduced basis element method (SCRBE) introduced in [5]. Rather than the eigenmodal expansion typically used in the CMS, the SCRBE employs the reduced basis method (RB) [6] for the bubble function approximations. Each RB approximation space is specifically tailored to a particular bubble and the associated parameter dependence defined by the PDE within each component; the SCRBE thus accommodates parametric variations for example related to component geometry, loads, material properties, or boundary conditions. Furthermore, thanks to the typically very rapid (often exponential) convergence of the RB approximation [7, 8], these RB spaces are low-dimensional and thus bubble function approximation is computationally inexpensive.
In addition to enabling parametric variations, the SCRBE features a strict offline-online computational decoupling. In the offline stage, the RB spaces and associated datasets for each component archetype in a component library is computed and stored. This stage requires FE solves and may thus be relatively expensive, but is carried out only once as a library preprocessing step. In the subsequent online stage, the user may instantiate any of the interoperable library archetypes, and assign to each component instantiation the desired parameter values; the RB bubble function approximations are then computed, and the Schur complement system is assembled and solved. This online step is much less expensive and in particular does never invoke the underlying FE discretization.
However, common to all these static-condensation-based approaches — including the SCRBE — is a global Schur complement linear-algebraic system of size equal to the total number of degrees of freedom associated with ports. For large systems with many components and ports, and in particular for problems with three-dimensional vector-valued field variables — such as in linear elasticity — the size of this system is considerable and thus clearly prohibits the fast response required in, say, an interactive design or optimization context. To overcome this limitation various port reduction techniques may be used. For example, for the CMS approaches an eigenmode expansion (with subsequent truncation) for the port degrees of freedom is considered in [9, 10], and an adaptive procedure based on a posteriori error estimators for the port reduction is considered in [11]. For the SCRBE, we introduce in [12] port reduction with empirical modes; in this case the port approximation spaces are informed by snapshots of relevant port-restricted solutions which are obtained through an offline pairwise empirical training algorithm.
Unique to the SCRBE is a certification framework that allows efficient computation of a posteriori bounds or estimators for the error in the SCRBE approximation with respect to the underlying FE “truth” discretization. This framework invokes classical residual arguments on the (RB) bubble level [6], a non-conforming approximation to the error-residual equation at the port level, and finally matrix perturbation at the system level in order to bound (under an eigenvalue proximity assumption) the error contributions from both RB approximation [5] and port reduction [12]. In actual practice, we may reduce online computational cost by consideration of a plausible and asymptotically rigorous error estimator rather than a rigorous error bound.
In this paper, we extend our earlier work for two-dimensional scalar problems in [12] to the more demanding three-dimensional vector-field case. We focus here on applications in linear elasticity, but we note that the component synthesis and indeed RB and port approximations can be readily extended to problems in heat transfer or (frequency domain) acoustics, or any phenomenon described by a linear elliptic or parabolic [13] PDE.a Through our procedure we effectively reduce the number of degrees of freedom from tens of millions (in the underlying FE discretization) to only a few thousand (in the port-reduced SCRBE approximation); the associated computation time is thus reduced from minutes or hours to only a few seconds.
Our approach here features several important innovations. First, as we consider here larger global systems with a much larger number of instantiated components we introduce a new non-symmetric SCRBE approximation, which reduces both offline and online cost and memory footprint; the corresponding linear-algebraic system is subsequently symmetrized in order to (say) accommodate efficient linear solvers. We also demonstrate that our central theoretical results in particular related to a posteriori error estimation survive intact for this more efficient revision of our earlier formulations in [12]. Second, we provide a precise formulation for general geometric mappings and port space compatibility, and we demonstrate that (in the isotropic linear-elastic case) rigid-body parameters related to “docking” of component instantiations in a system do not affect the associated bilinear forms and thus do not impact offline — thanks to smaller RB space dimensions — or online — thanks to treatment of differently oriented component instantiations as effectively identical — computational cost. Third, we introduce a new functional interpretation of our algebraic a posteriori error estimation framework in [12], which may serve to extend our approach here to larger classes of problems. And finally, we consider multi-reference parameter bound conditioners [14] for sharper error estimation.
The remainder of the paper is organized as follows. We start with a brief presentation of a general parametrized component static condensation framework for d-dimensional vector-valued linear elliptic partial differential equations; we focus on the concepts relevant in the SCRBE framework and we formulate the port compatibility requirements. Next, we discuss the RB and port reduction strategies for the computational cost reduction associated with component interiors and component interfaces, respectively. Then, we introduce our a posteriori error estimation framework. Finally, we present extensive results for a three-dimensional microtruss application, and provide some conclusive remarks.
Additional file 1: A short video which illustrates the methodology of this paper is published together with this paper as prscrbe_movie.mp4. (MP4 19 MB)
Component-based static condensation
Concepts: library components and system
We now introduce the key concepts for our SCRBE approximation: a library of parametrized and interoperable archetype components, which is prepared in the offline stage; and a system of component instantiations connected at ports, which is assembled and solved (and, if desired, visualized) in the online stage.
In the context of structural analysis, an archetype component typically (but not necessarily) corresponds to a physical construction unit, such as a beam, a plate, or a connector; in physical d-dimensional space (d=1,2,3) we denote by the reference domain associated with archetype component m, 1≤m≤M, where M is the number of archetypes in the library. The boundary of this domain, , has a set of disjoint local ports, denoted as , ; these ports enable the components to connect to other components. Note we shall assume that all ports on an archetype component are mutually separated by (at least) a non-port, non-Dirichlet boundary segment. If this is not the case, modifications to our procedures below must be considered [10].
where the and are parameter-independent forms and the and are parameter-dependent functions; for computational efficiency of the SCRBE evaluation stage it is critical that and are relatively small.
For simplicity of presentation here we shall assume that Dirichlet conditions are enforced only on ports and thus not through the archetype component discrete spaces (this is the case for our numerical results later).
An archetype component in coordinates .
Two component instantiations form a system in coordinates ( x,y ).
Note that here and in the following the notation denotes the usual composition,b and the notation denotes pointwise application of to the (vector-valued) argument;c we apply to the dependent variables to eliminate parameters related to spatial orientation of components from the bilinear forms, and to accommodate compatibility of basis functions on instantiated ports.
We may now introduce the synthesized system domain Ω as , the system parameter domain , and the system parameter vector μ=(μ1,…,μ I ); we denote the total number of system parameters by P.
When an instantiated component becomes part of a system, its local ports are associated to global ports. Each global port Γ p , , in the system is either a coincidence of two local ports and hence in the interior of Ω, or a single local port on the boundary ∂ Ω. We define the connectivity of the system through global-to-local index sets π p , : an interior global port is associated to two local ports γi,j and , and we thus set π p ={(i,j),(i′,j′)}; a boundary global port is associated to a single local port γi,j, and we thus set π p ={(i,j)}. We also introduce for instantiated component i, 1≤i≤I, a local-to-global map such that for local port j, , we have if (i,j)∈π p . Note that on any global port Γ p , , we may elect to impose Dirichlet boundary conditions; we denote by the number of global ports on which we do not impose Dirichlet boundary conditions.
we discuss this port compatibility requirement further in the “Port compatibility” subsection below.
note that the effect of the mapping to each archetype bilinear and linear form (defined over the archetype reference domain) is reflected through the parameter μ i .
In the case that is a pure rigid-body transformation (that is, is a rotation and a translation) and the material properties of the component do not depend on spatial orientation — such as in isotropic linear elasticity — the application of to the dependent variables results in cancellation of the mapping Jacobians, and thus the archetype bilinear form does not reflect the associated mapping parameters. Similarly, when is a combination of a rigid-body map and (say) dilation, only the latter must be parametrized through the archetype bilinear form. We explicitly demonstrate this cancellation for the case of isotropic linear elasticity in the “Microtruss beam application” section, and we comment on the computational implications in the “Model reduction” section.
we also introduce a compliance output as s(μ)=f(u(μ);μ). (Note that, as discussed in [5], restrictions apply to the geometric maps to maintain well-posedness of (10).)
we also introduce the FE compliance output s h (μ)=f(u h (μ);μ).
Mathematical formulation: static condensation
To formulate the static condensation procedure we decompose our discrete global space X h (Ω) into bubble spaces associated with component-interior degrees of freedom and a skeleton space associated with port degrees of freedom.
Note that is a continuous space thanks to the port compatibility requirement (7). Also note that in the definition of we include only the ports on which we do not impose Dirichlet boundary conditions (we assume without loss of generality that we enforce Dirichlet boundary conditions on global ports ).
Note that each is the fundamental solution (local to a component pair) of our (homogeneous) global equation associated with the particular port mode χi,j,k. Also note that and scale linearly with certain “free” parameters, such as component-wide thermal conductivity or Young’s modulus, which enter outside the bilinear form in (18) and (19); this will have important cost-saving implications in the context of RB approximation.
as before, our FE compliance output is s h (μ)=f(u h (μ);μ).
There is no distinction between (23) and (25) in the FE static condensation context; however in the context of the SCRBE, direct approximation of (23) leads to a non-symmetric Schur complement system, while direct approximation of (25) leads to a symmetric Schur complement system. In this paper we shall pursue the former with subsequent Schur complement symmetrization as the latter implies significantly larger online computational cost.
We shall invoke the interpretation (34) of to symmetrize the SCRBE Schur complement system below.
Port compatibility
The port compatibility requirement (7) between port basis functions associated with ports which may interconnect in a system — port of the same type — ensures solution continuity across shared global ports. We recall the archetype port basis functions introduced in (3), and we recall the associated physical (instantiated) port space basis functions χi,j,k introduced in (6). To honor (7), it is clear that the basis functions on different archetype ports of the same port type must be defined differently according to the archetype port orientation.
this map is the key to honor the port compatibility requirement (7).e
Note that .
We recall that (for 1≤i≤I) when applied to a port corresponds to pure translation. As a result, application of the port mapping corresponds only to translation and rotation. We now recall that the rotation applied to β on each side of (43) is unique, and we may thus conclude from (43) that . With (41) and (42), we then obtain , and we thus honor our port compatibility requirement (7).
Model reduction
The computational efficacy of our port-reduced SCRBE approach is realized through two separate model reduction techniques. As in the standard SCRBE approach [5] we consider component-interior model reduction through RB approximation [6] of the source bubbles (18) and of the fundamental solutions (19) to reduce the cost of each of the many component-interior linear solves required to form the Schur complement system. In addition to RB approximation in the component interiors, we employ port reduction [12] with empirical port modes to reduce the number of degrees of freedom on the ports and thus the size of the Schur complement system. We now discuss each of these techniques in more detail.
Component-interior reduction
and thus . The purpose of these RB approximations is to allow for efficient formation of an approximation to the Schur complement system (54): each RB approximation or is associated with a rapidly convergent [7] RB space specifically tailored to the particular bubble and to the parameter dependence defined by the corresponding (archetype domain) PDE (18) or (19). All RB bubble spaces are thus different, and furthermore each space is typically of much lower dimension than the original FE spaces . As a consequence, the RB approximations to the solutions of (18) and (19) are obtained at significantly reduced computational cost with minimal compromise to solution accuracy. The RB method is now considered standard, and we refer the reader to [6] for all technical details relevant to the particular class of problems (linear elliptic) that we consider here.
in (47), or we may explicitly symmetrize by algebraic manipulation. The former option necessitates larger offline and online computational cost and storage, in fact, when compared to the latter, by a multiplicative factor equal to the number of RB basis functions.
The associated SCRBE compliance output approximation is .
Note that in actual practice, we assemble (54) through a direct-stiffness procedure from component-local matrix and vector blocks associated with and assembled for each of the I component instantiations; the procedure is described in detail in [5, 12]. The assembly of these component-local quantities constitutes the majority of online computational cost. However, we need only perform the assembly for each unique component instantiation, as identical (or “cloned”) components may share local matrices and vectors. We thus realize significant computational savings for systems which consist of instantiations of many component clones, such that we need only consider Ieff≪I effective component instantiations for this assembly proceedure.
There are two particularly important situations in which different component instantiations are effectively clones in the sense that the component-local matrix and vector blocks may still be re-used: First, matrix and vector blocks computed for component instantiations which differ only in spatial orientation are (in the case that material properties do not depend on spatial orientation, such as in isotropic linear elasticity) identical thanks to cancellation of the mapping Jacobians in the archetype domain bilinear form; second, “free” parameters such as component-wide thermal conductivity or Young’s modulus enter outside the bilinear forms in (18) and (19), and thus the associated matrix blocks will only differ by a scaling factor. As a result, we often obtain Ieff≪I in practice. We discuss this situation further under “Computational procedures” later in this section.
Port reduction
Framework
While the RB approximation is concerned with component-interior model reduction, we apply port reduction to reduce the number of degrees of freedom associated with component interfaces. For the port reduction procedure we shall consider on each global port Γ p only port modes as “Active” and thus contributing to the approximation; for substantial computational savings we require . We consider in this subsection the generic port reduction framework and in the next subsection our particular choice of port space basis functions which realizes .
The associated port-reduced SCRBE compliance output approximation is .
The purpose of port reduction is of course to reduce the size of the Schur complement system — and thus computational cost — while maintaining accuracy of the approximation. The size of the system (64), nA, is equal to the total number of active port modes in the system. In practice, we shall typically invoke only a few port degrees of freedom on each port such that nA≪nSC. A good choice for the port modes χi,j,k is key to the accuracy of the port-reduced SCRBE approximation, and is the focus of the next subsection.
Empirical port mode training
To ensure port compatibility we must for each port type develop our port basis on the associated reference port domain β as discussed under “Port compatibility” above. To this end we pursue a pairwise training algorithm that provides a port space tailored to the family of solutions associated with this port type. We shall develop bases for the full port spaces (6) and not merely the space spanned by “Active” modes; the remaining “Inactive” modes shall play a role in certification (for residual calculation), which we discuss further in the “Certification framework” section.
Our port spaces shall consist of three sets of modes. The first set of modes is explicitly specified and consists of the six modes associated with rigid-body motion.f We include these six modes for two reasons: first, it simplifies the procedure for specification of typical Dirichlet boundary conditions, and second, it ensures invertibility of the Schur complement operator associated with “Inactive” modes, which is a property we require for our non-conforming error estimation framework.
The second set of modes consists of the modes which shall be the outcome of our pairwise training algorithm. The third set of modes consists of singular Sturm-Liouville eigenmodes restricted to the orthogonal complement of the first empirical modes [12]. These modes serve to complete the discrete port space in a numerically stable fashion.
Recall that the total number of modes associated with the reference port β is . We consider here the case d=3 and thus ; each port mode , , has the form , where the number of degrees of freedom associated with each field component is .
for the two modes associated with flipping. Note these six modes are mutually (L2(β)) d -orthonormal. (If β is not the square β= [−0.5,0.5]2 we apply Gram-Schmidt orthonormalization to these first six modes to recover (L2(β)) d -orthonormality.)
The next port modes are the outcome of our pairwise empirical training algorithm. In this algorithm we exploit the fact that within any system, the solution on any global (shared, say) port is determined completely by the parameter values assigned to the pair of components sharing the port and the (typically relatively smooth) solution on all other ports associated with these two components. The purpose of our pairwise training algorithm is to explore the associated “solution manifold” induced by local parameter dependence and neighboring ports in a systematic fashion such that the empirical modes associated with each port type are tailored to all possible component connectivity and all admissible component parameter values.
For our empirical training algorithm we shall require discrete “Legendre polynomials” , such that the are the eigenvectors of a scalar singular Sturm-Liouville eigenproblem [16] over β ordered according to increasing eigenvalue; we shall also require a univariate random variable r with uniform density; and we introduce an algorithm tuning parameter γ>1 related to anticipated regularity. We then identify one or several pairs of components in the component library that may connect through a global port of the relevant port type β.
The empirical training procedure for each such pair is now given by Algorithm 1: we sample (solve) each pair Nsample times for different (random) parameters and different (random but smooth thanks to the parameter γ>1) boundary conditions on all non-connected ports (note that we assign random boundary conditions independently to each vector component); for each such sample we extract the solution on the shared port of the relevant type, map it to the reference port β, subtract from this mapped solution its orthogonal (L2(β)) d -projection onto each of the six rigid body modes , 1≤i≤6, and then finally include the result ζ in a snapshot set Spair associated with the current pair. Note that in Algorithm 1 refers to the vector (L2(β)) d inner product.
We then perform a data compression step: we invoke the proper orthogonal decomposition (POD) [17] (with respect to the vector (L2(β)) d inner product). The output from the POD procedure is a set of mutually (L2(β)) d -orthonormal POD modes which are also orthonormal to the six first modes , 1≤i≤6, related to rigid-body motion. We choose these POD modes as our next reference port basis functions , ; we typically observe rapid (often exponential) convergence [12] of these POD modes with respect to the input snapshot set Stype.
We refer to all first port modes as our empirical port modes. If is chosen such that , we now complete the discrete space with Sturm-Liouville singular eigenmodes restricted to the orthogonal complement space (of dimension ) as discussed in detail in [12].
We finally note that for our pairwise training approach we may employ the (non-port-reduced) SCRBE framework or we may use standard FE approximations. The computational cost associated with empirical training is not critical as the procedure is performed offline. For our numerical results in this paper we have used the non-port-reduced SCRBE framework to calculate empirical modes.
Computational procedures
The computational procedures associated with our port-reduced SCRBE approximation framework naturally decouple into an offline preprosessing stage and an online evaluation stage, and we now discuss each in more detail. Note we provide here only descriptions of each of the offline and online steps involved; for detailed online operation counts we refer to [12].
Offline
The offline stage is the preprosessing stage — performed only once — in which we construct and prepare the archetype component library. This stage consists of the following steps.
Off1. Empirical pairwise training by Algorithm 1. For each port type we sample pairs of components to obtain efficient port space basis functions , , associated with each reference port domain β. In the current implementation, we employ the non-port-reduced SCRBE [5] (rather than standard global FE) for the pairwise training.
Off2. RB space construction. For each archetype component m, 1≤m≤M, we must train different RB spaces to accommodate the RB approximations (44) and (45). Each construction of an RB space requires a number of component-local FE solves (each associated with an RB space basis function), and thus this step is potentially rather expensive, depending on the component spatial discretization and parametric complexity and in the bilinear and linear form expansions (1).
Note, however, that the construction of the RB approximation spaces (subsequent to port space construction) is embarrassingly parallel. Also note that we do not consider parameters for spatial orientation (because of the mapping Jacobian cancellations in the archetype domain formulation), and furthermore recall that components often have “free” parameters such as component-wide thermal conductivity or Young’s modulus, with which the solutions to (18) and (19) simply scale linearly. As a result, RB space dimensions are typically rather small (around ten basis functions often suffice for each RB space), and thus although this step typically dominates offline cost the computational effort is not onerous: typically a couple of CPU hours is required for each archetype component.
Off3. Online dataset preparation. For each archetype component we construct data to enable efficient assembly of the component-local Schur complement matrix and vector blocks in the subsequent online stage. The computation time depends stongly on component spatial discretization and parametric complexity, but is typically between minutes and hours (on a single CPU) for each component. The online dataset also contains all RB basis functions, which are required for online global field visualization, if desired.
Off4. Data loading. We finally read the online datasets (typically a few hundred Mb) for all library components into computer memory to prepare for the online stage.
Note that with our current implementation of the framework, since we employ the non-port-reduced SCRBE in step Off1 above, we must first perform a step Off0a (similar to Off2) and then a step Off0b (similar to Off3) in order to enable the necessary “online” pair evaluation in Off1.
Online
The online stage is the stage in which we instantiate archetype components, and assemble and solve our system. This stage consists of the following steps, which in the current implementation is performed on a single CPU.
On1. Component instantiation. Instantiate I components from the library, assign the relevant parameter values to each component, and connect components to other components through ports of the same type to form a system; this step is most easily effected through a graphical user interface [Additional file 1].
On2. Schur complement system formation. Perform component-local RB solves (of small RB dimension) associated with all “Active” degrees of freedom to obtain (RB coefficients for) the RB approximations and , assemble the associated matrix and vector blocks for each component, and assemble the Schur complement system (64) through a direct-stiffness procedure [5, 12].
(the symmetrization is performed on the component level) and the entries in the component-local vector blocks are of the form ψi,j,k;μ); the subscripts A refer to assembly of “Active” component matrices and vectors. However, thanks to an efficient construction-evaluation procedure [6], which relies on the affine operator expansions (1), only the RB coefficients associated with and are required for this assembly step. We emphasize in particular that the underlying component FE discretization is never invoked. We recall that parameters related to spatial orientation (component “docking”) do not appear in the (archetype) bilinear forms due to cancellation of the associated Jacobians (we demonstrate this for isotropic linear elasticity in the “Microtruss beam application”section); and moreover, certain parametric variations such as component-wide conductivity or Young’s modulus are “free” in the sense that they enter as scalars outside the bilinear forms in (18) and (19). As a consequence, matrix and vector blocks associated with different component instantiations are in practice often identical (in the context of “free” parameters up to a multiplicative constant). We may thus in typical systems often consider only Ieff≪I effectively different (or unique) component instantiations, for which we perform RB solves and assemble component-local matrices and vectors. The component-local matrices and vectors for the remaining I−Ieff component instantiations are then simply copies of the respective data from effectively identical components. This consideration of component “clones” together with the realization of “docking” parameter cancellation and “free” parameters contribute significantly to the modest computational cost associated with On2.g The typical computation time is a few seconds.
On3. Evaluate. Solve the “Active” Schur complement system, and evaluate any relevant derived quantities from the solution vector (for example a compliance output). The typical computation time is a few seconds.
The computational cost associated with this online stage is dominated by On2 (when Ieff is close to I) or On3 (when Ieff≪I). However, the offline and online stages above are only concerned with the port-reduced SCRBE approximation. We consider the computational procedures associated with a posteriori error estimation in the next section.
Certification framework
Our port-reduced SCRBE approximation is equipped with efficiently computable a posteriori error bounds and estimators that provide certificates for the error in the approximation with respect to the underlying global FE discretization. We employ in this paper the energy-norm and compliance output bound developed in [12], and we present the main ingredients and certain extensions below. We furthermore sharpen the bounds by consideration of a multi-reference parameter bound conditioner.
The error in our approximation derives from two sources: port reduction and RB approximation. Below we first address the error due to port reduction, that is to say, the case in which the error due to RB approximation is zero. In this case the error bound presentation simplifies significantly and in particular admits a pure functional interpretation. We then subsequently perturb the equivalent algebraic interpretation to provide a bound for the general case in which the error due to RB approximation is non-zero.
Port reduction error contribution
note that because of the source bubble terms bf;h(μ i ) in (71).
this error-residual relationship is the point of departure for our error bound development.
and thus . This first relaxation of (78) not only provides a bound on the energy of the error field, but also accommodates efficient bound calculation thanks to the non-conforming space .
Thanks to incorporation of the modes related to rigid-body motion in our port space bases (presuming on all global ports Γ p , 1≤p≤n Γ ) we expect in general (and for a particular system, we computationally verify) that (91) is well-posed; for the simpler class of problems with scalar-valued fields we demonstrate this well-posedness in [12]. The RB-error-free bound given in (90) (together with (91)) is the basis on which we in the next subsection extend our error estimation framework to the general case of non-zero RB errors and furthermore to certain outputs of interest.
note that, thanks to (92) and the fact that , (95) is equivalent to (77).
for 1≤i,j≤nNC. Note that because of the Galerkin orthogonality in (19), and thus is indeed the non-conforming version of the Schur complement matrix in (26); similarly, note that because of (18) and the fact that vanish on ports, and thus is the non-conforming version of the vector in (26).
note that .
We note that the bound (107) necessitates a solve of dimension nNC≥nSC. However, this solve may be performed efficiently thanks to i) the non-conforming skeleton space which in a natural way allows component-local elimination of all degrees of freedom that do not couple at shared global ports; and ii) the quasi parameter-independent bound conditioner matrix associated with the bilinear form b μ , which allows offline pre-factorization for all these component-local solves. And furthermore, in actual practice we invoke not λmin(μ) but rather a computationally tractable eigenvalue lower bound . We consider computational aspects of our error estimation framework in more detail in the “Computational procedures” subsection below.
RB error contribution — A Posteriori error estimators
We now modify (107) in order to obtain an efficiently computable a posteriori error bound which is also valid in the presence of RB error contributions. First, as we in the SCRBE context only have access to an approximation of the FE Schur complement system, the residual can not be computed exactly and we thus instead compute a residual approximation together with bounds on associated RB-error-induced residual perturbation terms. Second, we introduce a lower bound (valid under an eigenvalue proximity assumption) for the eigenvalue λmin(μ) which is based on the solution to a port-reduced eigenproblem, an approximate eigenproblem residual, and bounds on associated RB-error-induced eigenproblem residual perturbation terms.
Moreover, in the presence of RB error contributions the error in the Schur energy is not equal to the energy of the error in the field, and thus in addition to a bound on the former we require a bound on additional RB perturbation terms to obtain a bound for the latter. Further, we develop in this section, from our Schur energy error bound, a new bound on port-restricted compliance outputs. For this output bound we must take into account that we in this paper (in contrast to in [12]) employ rather than (the former being a port-reduced version of the latter, which is defined in (51)) as our skeleton space. Finally, we introduce asymptotically rigorous error estimators, by which we reduce computational cost by neglecting typically very small quadratic RB error bound contributions.
We first develop a bound for the error in the Schur energy norm, , through perturbations of the left-hand side of (107). We subsequently modify this bound to obtain a bound on ∥e h (μ)∥ μ ; note the former is not equivalent to the latter because e h (μ) is not a member of .
To bound the error in the Schur energy, we must thus, based on residual and eigenvalue approximations, develop upper and lower bounds for the numerator and denominator, respectively, of the left-hand side of (115).
We now obtain a computable eigenvalue lower bound in
Lemma 1
Proof
We refer to ([12], Proposition 1) for the proof, and we note that a similar residual-based eigenvalue bound has been developed in [18] for the standard eigenproblem. □
With the residual approximation , associated RB error bounds σ(μ), and the eigenvalue lower bound λmin,LB(μ;C) above, we may now obtain a computable bound for the left-hand side of (115) and thus the error in the Schur energy norm in
Proposition 1
Proof
We merely note here that the numerator in (128) is an upper bound for the numerator in (115), and that λmin,LB(μ;C)≤λmin(μ) is a lower bound for the denominator in (115). We refer to ([12], Appendix A) for the detailed proof. □
We then introduce our bound for the energy of the error field in
Proposition 2
Proof
We refer to ([12], Appendix A) for the proof. □
We then state
Proposition 3
Proof
We provide here a full proof as in the present paper (skeleton space ) the proof is different from a related proof in [12] (skeleton space ).