Reduced Basis Element Research Articles

Constraints of local stiffness adaption for reduced-order models of large-scale steel bridges

Large and complex structures, such as bridges, are subjected to a variety of environmental factors and loads over time, leading to evolving structural states. To assess their current condition and plan predictive maintenance, Digital Twins enriched with Structural Health Monitoring (SHM) data are essential. However, for large-scale structures, the creation of physics-based Digital Twins necessitates reduced-order modeling to ensure efficient adaptation to SHM data. Many of these methods for structural damage assessment rely on localized stiffness reduction to characterize discrete damage states. The Static Condensation Reduced Basis Element (SCRBE) method is a promising approach for model order reduction by decomposing the structure into parametric components. However, SCRBE's effectiveness and the accuracy of the Digital Twin depend on an appropriate partitioning of the structure into to components with reduced stiffness. This paper systematically investigates the process and the constraints of component selection for steel bridges with discrete damages, considering factors like numerics, sensor placement and complex geometries. A comprehensive analysis is conducted using a real bridge case study augmented with synthetic measurement data representing damage scenarios. The results reveal both challenges and the potential of the SCRBE method for Digital Twins. A systematic approach for component selection within the SCRBE framework is proposed, specifically tailored for real-world application scenarios. This structured methodology facilitates the efficient and adaptive development of Digital Twins for large, intricate structures, such as steel bridges.

Lego-like composite laminate construction and analysis on the fly

High-fidelity finite element (FE) analysis is computational demanding for parametric studies and design optimizations of composite laminates because it is time-consuming and inflexible in reconfiguration. Hence, we aim to develop rapid yet accurate 3-D FE analysis that can readily change (i) layer materials including fiber angles, (ii) the number of layers, and (iii) the stacking sequence of layers. To achieve these three features, we propose an offline-online framework hinging on static condensation (SC) integrated with port reduction (PR) and reduced basis element (RBE) methods, resulting in port-reduced static condensation (PR-SC) and port-reduced static condensation reduced basis element (PR-SCRBE) analyses. Thanks to the PR-SC and PR-SCRBE techniques, we may easily build a composite laminate using a variety of layer materials and any desired number of layers, akin to assembling Lego bricks, and swiftly analyze the constructed composite laminate. For demonstration, we assess both PR-SC and PR-SCRBE approximations using three different composite laminates with varying numbers of layers for each of three-point bending and tensile tests. Compared with FE analysis, PR-SC analysis exhibits insignificant accuracy loss in displacement and stress evaluation while obtaining meaningful computational speedup. In contrast, PR-SCRBE analysis shows moderate accuracy loss in displacement and stress evaluation while gaining considerable computational speedup. In conclusion, we recommend employing PR-SC for accuracy and PR-SCRBE for efficiency as per application requirements. The proposed two-stage framework may benefit many-query and real-time applications involving composite laminate analysis, including design optimization and virtual testing.

Component-based wing structure reconfiguration and analysis on the fly

To facilitate the structural design of an aircraft wing, we propose a simulation technique that can not only readily reshape a wing structure, but also rapidly yet accurately analyze its responses. For swift wing reconfiguration, we express a wing geometry with respect to shape parameters, such as a taper ratio, a half wingspan, and a sweep angle, by geometric parameterization. For rapid yet accurate analysis, we simulate a large-sized wing structure using small-sized common components whose solution spaces are shrunken by port-reduced static condensation reduced basis element (PR-SCRBE) approximation. As a result, we can easily modify both the shape and topology of a wing structure in the context of repetitive analysis. For demonstration, we construct six exemplary wing structures by assembling deformed duplicates of five common wing components. With the six wing structures, we examine the accuracy and efficiency of linear elastostatic PR-SCRBE analyses in predicting displacements and stresses compared with the corresponding finite element (FE) analyses. Regarding accuracy, the six PR-SCRBE analyses yield displacements and stresses with average relative errors of less than 0.01% and 2.79%, respectively. As for efficiency, the six PR-SCRBE analyses evaluate displacements and stresses with average speedups of 12.58 and 11.56, respectively. In conclusion, the proposed approach for wing reconfiguration and analysis may produce rapid yet accurate structural predictions, thereby considerably assisting the structural design of an aircraft wing.

A reduced-basis element method for pin-by-pin reactor core calculations in diffusion and [formula omitted] approximations

The reduced order model (ROM) methods allow to significantly improve the computation time and memory usage. Therefore these methods are very useful for real-time or many-query reactor core simulation analysis. The ROM approach is based on the proper orthogonal decomposition methods which generate an optimal truncated subspace from a given set of test solutions or snapshots. In the case of high-fidelity real-size problems the calculation of one snapshot might be too expensive or not possible due to memory limitations. To circumvent this problem we apply the reduced basis element method (RBEM). This method is based on a spatial decomposition of the core domain and on the application of the ROM approach on each subdomain. In this work the RBEM is developed for diffusion and SP3 equations and the results are illustrated with several two-dimensional reactor core calculations.

Component Mapping Automation for Parametric Component Reduced Basis Techniques (RB-COMPONENT)

The aim of this paper is to develop some techniques for automation of the mappings (between working and reference domains) required by reduced basis methods: the development of geometry mappings is indeed often a substantial impediment to the implementation of reduced basis techniques, especially in the context of the reduced basis element method (RBEM) and the reduced basis component method (RBCM). In the RBCM context, the geometry mappings are applied at the level of components. The methods have been tested on various cases to understand the limits of the approach and try to foresee and overcome the possible failures.

Proper orthogonal decomposition‐based reduced basis element thermal modeling of integrated circuits

SummaryVarious reduced basis element methods are compared for performing transient thermal simulations of integrated circuits. The reduced basis element method is a type of reduced order modeling that takes advantage of repeated geometrical features. It uses a reduced set of basis functions to approximate the solution of a PDE on subdomains (blocks); then these blocks are coupled together to perform a simulation on an entire domain. As the simulations are transient, a proper orthogonal decomposition basis is used, and the proper orthogonal decomposition eigenvalues from each block are used to derive error bounds for the entire simulation. These bounds are used to examine choices of block decompositions for a simplified integrated circuit structure. A decomposition that uses a single block for each transistor device is compared with a decomposition that uses one block for multiple devices. It was found that larger blocks are more computationally efficient; however, the advantage decreases if the devices within a block receive independent signals. Continuous and discontinuous methods of coupling the blocks were also compared. The coupling methods lend themselves to different solution approaches such as static condensation (continuous coupling) and block‐based inversion (discontinuous). Static condensation yielded the best convergence rate, accuracy, and operation count. Copyright © 2017 John Wiley & Sons, Ltd.

A Static Condensation Reduced Basis Element Approach for the Reynolds Lubrication Equation

AbstractIn this paper, we propose a Static Condensation Reduced Basis Element (SCRBE) approach for the Reynolds Lubrication Equation (RLE). The SCRBE method is a computational tool that allows to efficiently analyze parametrized structures which can be decomposed into a large number of similar components. Here, we extend the methodology to allow for a more general domain decomposition, a typical example being a checkerboard-pattern assembled from similar components. To this end, we extend the formulation and associateda posteriorierror bound procedure. Our motivation comes from the analysis of the pressure distribution in plain journal bearings governed by the RLE. However, the SCRBE approach presented is not limited to bearings and the RLE, but directly extends to other component-based systems. We show numerical results for plain bearings to demonstrate the validity of the proposed approach.

Spectral based Discontinuous Galerkin Reduced Basis Element method for parametrized Stokes problems

In this work we extend to the Stokes problem the Discontinuous Galerkin Reduced Basis Element (DGRBE) method introduced in Antonietti et al. (2015). By this method we aim at reducing the computational cost for the approximation of a parametrized Stokes problem on a domain partitioned into subdomains. During an offline stage, expensive but performed only once, a low-dimensional approximation space is built on each subdomain. For any new value of the parameter, the rapid evaluation of the solution takes place during the online stage and consists in a Galerkin projection onto the low-dimensional subspaces computed offline. The high-fidelity discretization on each subdomain, used to build the local low-dimensional subspaces, is based on spectral element methods. The continuity of both the velocity and the normal component of the Cauchy stress tensor at subdomain interfaces is weakly enforced by a discontinuous Galerkin approach.

A discontinuous Galerkin reduced basis element method for elliptic problems

We propose and analyse a new discontinuous reduced basis element method for the approximation of parametrized elliptic PDEs in partitioned domains. The method is built upon an offline stage (parameter independent) and an online (parameter dependent) one. In the offline stage we build a non-conforming (discontinuous) global reduced space as a direct sum of local basis functions generated independently on each subdomain. In the online stage, for any given value of the parameter, the approximate solution is obtained by ensuring the weak continuity of the fluxes and of the solution itself thanks to a discontinuous Galerkin approach. The new method extends and generalizes the methods introduced in [L. Iapichino, Ph.D. thesis, EPF Lausanne (2012); L. Iapichino, A. Quarteroni and G. Rozza, Comput. Methods Appl. Mech. Eng. 221–222 (2012) 63–82]. We prove its stability and convergence properties, as well as the spectral properties of the associated online algebraic system. We also propose a two-level preconditioner for the online problem which exploits the pre-existing decomposition of the domain and is based upon the introduction of a global coarse finite element space. Numerical tests are performed to verify our theoretical results.

Component-based reduced basis for parametrized symmetric eigenproblems

A component-based approach is introduced for fast and flexible solution of parameter-dependent symmetric eigenproblems. Considering a generalized eigenproblem with symmetric stiffness and mass operators, we start by introducing a “ σ-shifted” eigenproblem where the left hand side operator corresponds to an equilibrium between the stiffness operator and a weighted mass operator, with weight-parameter σ>0. Assuming that σ=λ n >0, the nth real positive eigenvalue of the original eigenproblem, then the shifted eigenproblem reduces to the solution of a homogeneous linear problem. In this context, we can apply the static condensation reduced basis element (SCRBE) method, a domain synthesis approach with reduced basis (RB) approximation at the intradomain level to populate a Schur complement at the interdomain level. In the Offline stage, for a library of archetype subdomains we train RB spaces for a family of linear problems; these linear problems correspond to various equilibriums between the stiffness operator and the weighted mass operator. In the Online stage we assemble instantiated subdomains and perform static condensation to obtain the “ σ-shifted” eigenproblem for the full system. We then perform a direct search to find the values of σ that yield singular systems, corresponding to the eigenvalues of the original eigenproblem. We provide eigenvalue a posteriori error estimators and we present various numerical results to demonstrate the accuracy, flexibility and computational efficiency of our approach. We are able to obtain large speed and memory improvements compared to a classical Finite Element Method (FEM), making our method very suitable for large models commonly considered in an engineering context.

A new certification framework for the port reduced static condensation reduced basis element method

In this paper we introduce a new certification framework for the port-reduced static condensation reduced basis element (PR-SCRBE) method, which has been developed for the simulation of large component based applications such as bridges or acoustic waveguides. In an offline computational stage we construct a library of interoperable parametrized reference components; in the subsequent online stage we instantiate and connect the components at the interfaces/ports to form a system of components. To compute a “truth” finite element approximation of the (say) coercive elliptic partial differential equation on the component based system we use a domain decomposition approach. For an efficient simulation we employ two different types of model reduction — a reduced basis (RB) approximation within the interior of the component (Huynh et al. 2013) and empirical port reduction (Eftang and Patera 2013) on the ports where the components connect.We demonstrate the well-posedness of the PR-SCRBE approximation and introduce a new certification framework. To assess the quality of the port reduction we use conservative fluxes. We adapt the standard estimators from RB methods to the SCRBE setting to derive an a posteriori error estimator for the RB-error contribution. In order to combine the a posteriori estimators for both error contributions and derive a rigorous a posteriori error estimator for PR-SCRBE we adapt techniques from multi-scale methods and component mode synthesis. Finally, we prove that the effectivity of the derived estimator can be bounded.We provide numerical experiments for heat conduction and linear elasticity to show that the derived a posteriori error estimator provides an effective estimator. Moreover we demonstrate the applicability of the introduced certification framework by analyzing the computational (online) costs.

A STATIC CONDENSATION REDUCED BASIS ELEMENT APPROXIMATION: APPLICATION TO THREE-DIMENSIONAL ACOUSTIC MUFFLER ANALYSIS

We apply the static condensation reduced basis element (scRBE) method to treat the class of parametrized complex Helmholtz partial differential equation. We construct a set of components of interoperable parametrized reference components in a Library to model a family of target models relevant to acoustic devices. The components in the Library are built upon the scRBE method by using reduced basis (RB) formulation, and are compatible to each other by a set of common interfaces, or port. We apply an offline–online computational strategy to achieve rapid and accurate prediction of any parametric systems formed from a set of components in a Library. We demonstrate that the approach can handle large scale models with many parameters and/or topology variations efficiency in several numerical examples. We show that significant computational savings can be obtained by the scRBE method.

A port-reduced static condensation reduced basis element method for large component-synthesized structures: approximation and A Posteriori error estimation

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. 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. 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.

The Static Condensation Reduced Basis Element Method for a Mixed-Mean Conjugate Heat Exchanger Model

We propose a new approach for the simulation of conjugate heat exchangers. First, we introduce a dimensionality-reduced mathematical model for conjugate (fluid-solid) heat transfer: in the fluid channels, we consider a mixed-mean temperature defined on one-dimensional filaments; in the solid we consider a detailed partial differential equation conduction representation. We then propose a Petrov--Galerkin finite element (FE) numerical approximation which provides suitable stability and accuracy for our mathematical model. We next apply the static condensation reduced basis element (scRBE) method: a domain synthesis approach with parametric model order reduction at the intradomain level to populate a Schur complement at the interdomain level. We first build a library of “components,” each corresponding to a subdomain with a simple fluid channel geometry; for each component, we prepare Petrov--Galerkin reduced basis bubble approximations (and error bounds). We then assemble the system equations by static condensation and solve for the temperature distribution in the full domain. System-level error bounds are derived from matrix perturbation arguments; we also introduce a new output error bound which is sharper than the original scRBE estimator. We present numerical results for a two-dimensional automotive radiator model which demonstrate the flexibility, accuracy, and computational efficiency of our approach.

Port reduction in parametrized component static condensation: approximation and a posteriori error estimation

SUMMARYWe introduce a port (interface) approximation and a posteriori error bound framework for a general component‐based static condensation method in the context of parameter‐dependent linear elliptic partial differential equations. The key ingredients are as follows: (i) efficient empirical port approximation spaces—the dimensions of these spaces may be chosen small to reduce the computational cost associated with formation and solution of the static condensation system; and (ii) a computationally tractable a posteriori error bound realized through a non‐conforming approximation and associated conditioner—the error in the global system approximation, or in a scalar output quantity, may be bounded relatively sharply with respect to the underlying finite element discretization.Our approximation and a posteriori error bound framework is of particular computational relevance for the static condensation reduced basis element (SCRBE) method. We provide several numerical examples within the SCRBE context, which serve to demonstrate the convergence rate of our port approximation procedure as well as the efficacy of our port reduction error bounds. Copyright © 2013 John Wiley & Sons, Ltd.

A static condensation reduced basis element method: Complex problems

We extend the static condensation reduced basis element (scRBE) method to treat the class of parametrized complex Helmholtz partial differential equations. The main ingredients are (i) static condensation at the interdomain level, (ii) a conforming eigenfunction “port” representation at the interface level, (iii) the reduced basis (RB) approximation of finite element (FE) bubble functions at the intradomain level, and (iv) rigorous system-level error bounds which reflect RB perturbation of the FE Schur complement. We then incorporate these ingredients in an Offline–Online computational strategy to achieve rapid and accurate prediction of parametric systems formed from instantiations of interoperable parametrized archetype components from a Library. We introduce a number of extensions with respect to the original scRBE framework: first, primal–dual RB methods for general non-symmetric (complex) problems; second, stability constant procedures for weakly coercive problems (at both the interdomain level and intradomain level); third, treatment of non-port linear–functional outputs (as well as functions of outputs); fourth, consideration of particular components and outputs relevant to acoustic applications. We consider several numerical examples in acoustics (in particular focused on mufflers and horns) to demonstrate that the approach can handle models with many parameters and/or topology variations with particular reference to waveguide-like applications.

Towards beating the curse of dimensionality for gravitational waves using reduced basis

Using the reduced basis approach, we efficiently compress and accurately represent the space of waveforms for nonprecessing binary black hole inspirals, which constitutes a four-dimensional parameter space (two masses, two spin magnitudes). Compared to the nonspinning case, we find that only a marginal increase in the (already relatively small) number of reduced basis elements is required to represent any nonprecessing waveform to nearly numerical round-off precision. Most parameters selected by the algorithm are near the boundary of the parameter space, leaving the bulk of its volume sparse. Our results suggest that the full eight-dimensional space (two masses, two spin magnitudes, four spin orientation angles on the unit sphere) may be highly compressible and represented with very high accuracy by a remarkably small number of waveforms, thus providing some hope that the number of numerical relativity simulations of binary black hole coalescences needed to represent the entire space of configurations is not intractable. Finally, we find that the distribution of selected parameters is robust to different choices of seed values starting the algorithm, a property which should be useful for indicating parameters for numerical relativity simulations of binary black holes. In particular, we find that the mass ratios ${m}_{1}/{m}_{2}$ of nonspinning binaries selected by the algorithm are mostly in the interval $[1,3]$ and that the median of the distribution follows a power-law behavior $\ensuremath{\sim}({m}_{1}/{m}_{2}{)}^{\ensuremath{-}5.25}$.

A reduced basis hybrid method for the coupling of parametrized domains represented by fluidic networks

In this paper we propose a reduced basis hybrid method (RBHM) for the approximation of partial differential equations in domains represented by complex networks where topological features are recurrent. The RBHM is applied to Stokes equations in domains which are decomposable into smaller similar blocks that are properly coupled.The RBHM is built upon the reduced basis element method (RBEM) and it takes advantage from both the reduced basis methods (RB) and the domain decomposition method. We move from the consideration that the blocks composing the computational domain are topologically similar to a few reference shapes. On the latter, representative solutions, corresponding to the same governing partial differential equations, are computed for different values of some parameters of interest, representing, for example, the deformation of the blocks. A generalized transfinite mapping is used in order to produce a global map from the reference shapes of each block to any deformed configuration.The desired solution on the given original computational domain is recovered as projection of the previously precomputed solutions and then glued across subdomain interfaces by suitable coupling conditions.The geometrical parametrization of the domain, by transfinite mapping, induces non-affine parameter dependence: an empirical interpolation technique is used to recover an approximate affine parameter dependence and a subsequent offline/online decomposition of the reduced basis procedure. This computational decomposition yields a considerable reduction of the problem complexity. Results computed on some combinations of 2D and 3D geometries representing cardiovascular networks show the advantage of the method in terms of reduced computational costs and the quality of the coupling to guarantee continuity of both stresses, pressure and velocity at subdomain interfaces.

GLOBAL C1 MAPS ON GENERAL DOMAINS

In many contexts, there is a need to construct C1 maps from a given reference domain to a family of deformed domains. In our case, the motivation comes from the application of the Arbitrary Lagrangian Eulerian (ALE) method and also the reduced basis element method. In these methods, the maps are used to construct the grid-points needed on the deformed domains, and the corresponding Jacobian of the map is used to map vector fields from one domain to another. In order to keep the continuity of the mapped vector fields, the Jacobian must be continuous, and thus the maps need to be C1. In addition, the constructed grids on the deformed domains should be quality grids in the sense that, for a given partial differential equation defined on any of the deformed domains, the solution should be accurate. Since we are interested in a family of deformed domains, we consider the solutions of the partial differential equation to be a family of solutions governed by the geometry of the domains. Different mapping strategies are discussed and compared: the transfinite interpolation proposed by Gordon and Hall,12 the pseudo-harmonic extension proposed by Gordon and Wixom,13 a new generalization of the Gordon–Hall method (e.g., to general polygons in two dimensions), the harmonic extension, and the mean-valued extension proposed by Floater.8

A reduced basis element method for the steady Stokes problem

The reduced basis element method is a new approach for approximating the solution of problems described by partial differential equations. The method takes its roots in domain decomposi- tion methods and reduced basis discretizations. The basic idea is to first decompose the computational domain into a series of subdomains that are deformations of a few reference domains (or generic com- putational parts). Associated with each reference domain are precomputed solutions corresponding to the same governing partial differential equation, but solved for different choices of deformations of the reference subdomains and mapped onto the reference shape. The approximation corresponding to a new shape is then taken to be a linear combination of the precomputed solutions, mapped from the generic computational part to the actual computational part. We extend earlier work in this direction to solve incompressible fluid flow problems governed by the steady Stokes equations. Particular focus is given to satisfying the inf-sup condition, to a posteriori error estimation, and to gluing the local solutions together in the multidomain case.