A Static condensation Reduced Basis Element method: approximation and a posteriori error estimation

We propose a new reduced basis element-cum-component mode synthesis approach for parametrized elliptic coercive partial differential equations. In the Offline stage we construct a Library of interoperable parametrized reference components relevant to some family of problems; in the Online stage we instantiate and connect reference components (at ports) to rapidly form and query parametric systems. The method is based on static condensation at the interdomain level, a conforming eigenfunction “port” representation at the interface level, and finally Reduced Basis (RB) approximation of Finite Element (FE) bubble functions at the intradomain level. We show under suitable hypotheses that the RB Schur complement is close to the FE Schur complement: we can thus demonstrate the stability of the discrete equations; furthermore, we can develop inexpensive and rigorous (system-level) a posteriori error bounds. We present numerical results for model many-parameter heat transfer and elasticity problems with particular emphasis on the Online stage; we discuss flexibility, accuracy, computational performance, and also the effectivity of the a posteriori error bounds.United States. Air Force Office of Scientific Research. Multidisciplinary University Research Initiative (Grant FA9550-09-1-0613)MIT-Singapore International Design Cente

A Laplace transform certified reduced basis method; application to the heat equation and wave equation

We present a certified reduced basis (RB) method for the heat equation and wave equation. The critical ingredients are certified RB approximation of the Laplace transform; the inverse Laplace transform to develop the time-domain RB output approximation and rigorous error bound; a (Butterworth) filter in time to effect the necessary “modal” truncation; RB eigenfunction decomposition and contour integration for Offline–Online decomposition. We present numerical results to demonstrate the accuracy and efficiency of the approach.United States. Air Force Office of Scientific Research (Grant FA9550-07-1-0425)United States. Air Force Office of Scientific Research (Grant FA9550-09-1-0613

Certified reduced basis model validation: A frequentistic uncertainty framework

We introduce a frequentistic validation framework for assessment — acceptance or rejection — of the consistency of a proposed parametrized partial differential equation model with respect to (noisy) experimental data from a physical system. Our method builds upon the Hotelling T[superscript 2] statistical hypothesis test for bias first introduced by Balci and Sargent in 1984 and subsequently extended by McFarland and Mahadevan (2008). Our approach introduces two new elements: a spectral representation of the misfit which reduces the dimensionality and variance of the underlying multivariate Gaussian but without introduction of the usual regression assumptions; a certified (verified) reduced basis approximation — reduced order model — which greatly accelerates computational performance but without any loss of rigor relative to the full (finite element) discretization. We illustrate our approach with examples from heat transfer and acoustics, both based on synthetic data. We demonstrate that we can efficiently identify possibility regions that characterize parameter uncertainty; furthermore, in the case that the possibility region is empty, we can deduce the presence of “unmodeled physics” such as cracks or heterogeneities.United States. Air Force Office of Scientific Research. Multidisciplinary University Research Initiative (Grant FA9550-09-1-0613)MIT-Singapore International Design Cente

High-fidelity real-time simulation on deployed platforms

We present a certified reduced basis method for high-fidelity real-time solution of parametrized partial differential equations on deployed platforms. Applications include in situ parameter estimation, adaptive design and control, interactive synthesis and visualization, and individuated product specification. We emphasize a new hierarchical architecture particularly well suited to the reduced basis computational paradigm: the expensive Offline stage is conducted pre-deployment on a parallel supercomputer (in our examples, the TeraGrid machine Ranger); the inexpensive Online stage is conducted “in the field” on ubiquitous thin/inexpensive platforms such as laptops, tablets, smartphones (in our examples, the Nexus One Android-based phone), or embedded chips. We illustrate our approach with three examples: a two-dimensional Helmholtz acoustics “horn” problem; a three-dimensional transient heat conduction “Swiss Cheese” problem; and a three-dimensional unsteady incompressible Navier–Stokes low-Reynolds-number “eddy-promoter” problem.United States. Air Force Office of Scientific Research (Grant FA9550-07-1-0425)United States. Air Force Office of Scientific Research (OSD Grant FA9550-09-1-0613)National Science Foundation (U.S.) (University of Texas at Austin. Texas Advanced Computing Center Grant TG-ASC100016

Reduced Basis Approximation and A Posteriori Error Estimation for Stress Intensity Factors: Application to Failure Analysis

This paper reports the development of reduced basis approximations, rigorous a posteriori error bounds, and offline-online computational procedures for the accurate, fast and reliable predictions of stress intensity factors or strain energy release rate for “Mode I” linear elastic crack problem. We demonstrate the efficiency and rigor of our numerical method in several examples. We apply our method to a practical failure design application.Singapore-MIT Alliance (SMA

Real-Time Reliable Prediction of Linear-Elastic Mode-I Stress Intensity Factors for Failure Analysis

Modern engineering analysis requires accurate, reliable and efficient evaluation of outputs of interest. These outputs are functions of "input" parameter that serve to describe a particular configuration of the system, typical input geometry, material properties, or boundary conditions and loads. In many cases, the input-output relationship is a functional of the field variable - which is the solution to an input-parametrized partial differential equations (PDE). The reduced-basis approximation, adopting off-line/on-line computational procedures, allows us to compute accurate and reliable functional outputs of PDEs with rigorous error estimations. The operation count for the on-line stage depends only on a small number N and the parametric complexity of the problem, which make the reduced-basis approximation especially suitable for complex analysis such as optimizations and designs. In this work we focus on the development of finite-element and reduced-basis methodology for the accurate, fast, and reliable prediction of the stress intensity factors or strain-energy release rate of a mode-I linear elastic fracture problem. With the use of off-line/on-line computational strategy, the stress intensity factor for a particular problem can be obtained in miliseconds. The method opens a new promising prospect: not only are the numerical results obtained only in miliseconds with great savings in computational time; the results are also reliable - thanks to the rigorous and sharp a posteriori error bounds. The practical uses of our prediction are presented through several example problems.Singapore-MIT Alliance (SMA

Carbon nanotube four-terminal devices for pressure sensing applications

Carbon nanotubes (CNTs) are of high interest for sensing applications,owing to their superior mechanical strength, high Young’s modulus and low density. In this work, we report on a facile approach for the fabrication of carbon nanotube devices using a four terminal configuration. Oriented carbon nanotube films were pulled out from a CNT forest wafer and then twisted into a yarn. Both the CNT film and yarn were arranged on elastomer membranes/diaphragms which were arranged on a laser cut acrylic frame to form pressure sensors. The sensors were calibrated using a precisely controlled pressure system, showing a large change of the output voltage of approximately 50 mV at a constant supply current of 100 μA and under a low applied pressure of 15 mbar. The results indicate the high potential of using CNT films and yarns for pressure sensing applications

Magneto-transport properties of monolayer borophene in perpendicular magnetic field: influence of electron-phonon interaction

The magneto-transport properties of a borophene monolayer in a perpendicular magnetic field B are studied via calculating the conductivity tensor and resistance under electron-optical phonon interaction by using the linear response theory. Numerical results are obtained and discussed for some specific parameters. The magnetic field-dependent longitudinal conductivity shows the magneto-phonon resonance effect that describes the transition of electrons between Landau levels by absorbing/emitting an optical phonon. The Hall conductivity increases first and then decreases with the magnetic field strength. Also, the longitudinal resistance increases significantly with increasing temperature, which shows the metal behaviour of the material. Practically, the observed magneto-phonon resonance can be applied to experimentally determine some material parameters, such as the distance between Landau levels and the optical phonon energy

Component-based reduced basis for parametrized symmetric eigenproblems

Background: A component-based approach is introduced for fast and flexible solution of parameter-dependent symmetric eigenproblems. Methods: 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. Results: We provide eigenvalue a posteriori error estimators and we present various numerical results to demonstrate the accuracy, flexibility and computational efficiency of our approach. Conclusions: 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.United States. Air Force Office of Scientific Research (OSD/AFOSR/MURI Grant FA9550-09-1-0613)United States. Office of Naval Research (ONR Grant N00014-11-1-0713)Deshpande Center for Technological Innovation (grant)Switzerland. Commission for Technology and Innovation (CTI

A natural-norm Successive Constraint Method for inf-sup lower bounds

We present a new approach for the construction of lower bounds for the inf-sup stability constants required in a posteriori error analysis of reduced basis approximations to affinely parametrized partial differential equations. We combine the “linearized” inf-sup statement of the natural-norm approach with the approximation procedure of the Successive Constraint Method (SCM): the former (natural-norm) provides an economical parameter expansion and local concavity in parameter—a small(er) optimization problem which enjoys intrinsic lower bound properties; the latter (SCM) provides a systematic optimization framework—a Linear Program (LP) relaxation which readily incorporates continuity and stability constraints. The natural-norm SCM requires a parameter domain decomposition: we propose a greedy algorithm for selection of the SCM control points as well as adaptive construction of the optimal subdomains. The efficacy of the natural-norm SCM is illustrated through numerical results for two types of non-coercive problems: the Helmholtz equation (for acoustics, elasticity, and electromagnetics), and the convection–diffusion equation.United States. Air Force Office of Scientific Research (Grant No. FA 9550-07-1-0425