scholarly journals Real-Time Reliable Simulation of Heat Transfer Phenomena

2009 ◽  
Author(s):  
G. Rozza ◽  
D. B. P. Huynh ◽  
N. C. Nguyen ◽  
A. T. Patera
Keyword(s):  
Heat Transfer ◽  
Input Parameter ◽  
High Fidelity ◽  
Educational Context ◽  
Finite Element Discretization ◽  
Reduced Basis ◽  
Element Discretization ◽  
Class Project ◽  
Worked Problems ◽  
Basis Method

In this paper we discuss the application of the certified reduced basis method and the associated software package rbMIT© to “worked problems” in steady and unsteady conduction. Each worked problem is characterized by an input parameter vector — material properties, boundary conditions and sources, and geometry — and desired outputs — selected fluxes and temperatures. The methodology and associated rbMIT© software, as well as the educational worked problem framework, consists of two distinct stages: an Offline (or “Instructor”) stage in which a new heat transfer worked problem is first created; and an Online (or “Lecturer”/“Student”) stage in which the worked problem is subsequently invoked in (say) various in-class, project, or homework settings. In the very inexpensive Online stage, given an input parameter value, the software returns both (i) an accurate reduced basis output prediction, and (ii) a rigorous bound for the error in the reduced basis prediction relative to an underlying expensive high-fidelity finite element discretization; as required in the educational context, the response is both rapid and reliable. We present illustrative results for two worked problems: a steady thermal fin, and unsteady thermal analysis of a delamination crack.

Reduced Basis Methods and a Posteriori Error Estimators for Heat Transfer Problems

2009 ◽  
Author(s):  
G. Rozza ◽  
C. N. Nguyen ◽  
A. T. Patera ◽  
S. Deparis
Keyword(s):  
Natural Convection ◽  
Aspect Ratio ◽  
Reduced Basis Method ◽  
Reduced Basis ◽  
Element Discretization ◽  
Convection Problem ◽  
Prandtl Numbers ◽  
Basis Method ◽  
Two Parameters ◽  
Natural Convection Problem

This paper focuses on the parametric study of steady and unsteady forced and natural convection problems by the certified reduced basis method. These problems are characterized by an input-output relationship in which given an input parameter vector — material properties, boundary conditions and sources, and geometry — we would like to compute certain outputs of engineering interest — heat fluxes and average temperatures. The certified reduced basis method provides both (i) a very inexpensive yet accurate output prediction, and (ii) a rigorous bound for the error in the reduced basis prediction relative to an underlying expensive high-fidelity finite element discretization. The feasibility and efficiency of the method is demonstrated for three natural convection model problems: a scalar steady forced convection problem in a rectangular channel is characterized by two parameters — Pe´clet number and the aspect ratio of the channel — and an output — the average temperature over the domain; a steady natural convection problem in a laterally heated cavity is characterized by three parameters — Grashof and Prandtl numbers, and the aspect ratio of the cavity — and an output — the inverse of the Nusselt number; and an unsteady natural convection problem in a laterally heated cavity is characterized by two parameters — Grashof and Prandtl numbers — and a time-dependent output — the average of the horizontal velocity over a specified area of the cavity.

A Bi-Fidelity Approach for Uncertainty Quantification of Heat Transfer in a Rectangular Ribbed Channel

2016 ◽  
Author(s):  
Alireza Doostan ◽  
Gianluca Geraci ◽  
Gianluca Iaccarino
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Temperature Distribution ◽  
Rectangular Channel ◽  
Numerical Test ◽  
Computational Cost ◽  
High Fidelity ◽  
Reduced Basis ◽  
Ribbed Channel ◽  
Fidelity Model

This paper presents a bi-fidelity simulation approach to quantify the effect of uncertainty in the thermal boundary condition on the heat transfer in a ribbed channel. A numerical test case is designed where a random heat flux at the wall of a rectangular channel is applied to mimic the unknown temperature distribution in a realistic application. To predict the temperature distribution and the associated uncertainty over the channel wall, the fluid flow is simulated using 2D periodic steady Reynolds-Averaged Navier-Stokes (RANS) equations. The goal of this study is then to illustrate that the cost of propagating the heat flux uncertainty may be significantly reduced when two RANS models with different levels of fidelity, one low (cheap to simulate) and one high (expensive to evaluate), are used. The low-fidelity model is employed to learn a reduced basis and an interpolation rule that can be used, along with a small number of high-fidelity model evaluations, to approximate the high-fidelity solution at arbitrary samples of heat flux. Here, the low- and high-fidelity models are, respectively, the one-equation Spalart-Allmaras and the two-equation shear stress transport k–ω models. To further reduce the computational cost, the Spalart-Allmaras model is simulated on a coarser spatial grid and the non-linear solver is terminated prior to the solution convergence. It is illustrated that the proposed bi-fidelity strategy accurately approximates the target high-fidelity solution at randomly selected samples of the uncertain heat flux.

scholarly journals A space-time hp-interpolation-based certified reduced basis method for Burgers' equation

2014 ◽  
Vol 24 (09) ◽  
pp. 1903-1935 ◽  
Author(s):  
Masayuki Yano ◽  
Anthony T. Patera ◽  
Karsten Urban
Keyword(s):  
Error Bounds ◽  
Peclet Number ◽  
Burgers Equation ◽  
Space Time ◽  
Reduced Basis Method ◽  
Reduced Basis ◽  
Element Discretization ◽  
Péclet Number ◽  
Wide Range ◽  
Basis Method

We present a space-time interpolation-based certified reduced basis method for Burgers' equation over the spatial interval (0, 1) and the temporal interval (0, T] parametrized with respect to the Peclet number. We first introduce a Petrov–Galerkin space-time finite element discretization which enjoys a favorable inf–sup constant that decreases slowly with Peclet number and final time T. We then consider an hp interpolation-based space-time reduced basis approximation and associated Brezzi–Rappaz–Raviart a posteriori error bounds. We describe computational offline–online decomposition procedures for the three key ingredients of the error bounds: the dual norm of the residual, a lower bound for the inf–sup constant, and the space-time Sobolev embedding constant. Numerical results demonstrate that our space-time formulation provides improved stability constants compared to classical L2-error estimates; the error bounds remain sharp over a wide range of Peclet numbers and long integration times T, in marked contrast to the exponentially growing estimate of the classical formulation for high Peclet number cases.

Reduced Basis Approaches in Time-Dependent Non-Coercive Settings for Modelling the Movement of Nuclear Reactor Control Rods

2016 ◽  
Vol 20 (1) ◽  
pp. 23-59
Author(s):  
Alberto Sartori ◽  
Antonio Cammi ◽  
Lelio Luzzi ◽  
Gianluigi Rozza
Keyword(s):  
Nuclear Reactor ◽  
Time Dependent ◽  
Reduced Basis ◽  
Reactor Control ◽  
Element Discretization ◽  
Control Rod ◽  
Basis Method ◽  
And Control ◽  
Control Rods ◽  
Nuclear Reactor Control

AbstractIn this work, two approaches, based on the certified Reduced Basis method, have been developed for simulating the movement of nuclear reactor control rods, in time-dependent non-coercive settings featuring a 3D geometrical framework. In particular, in a first approach, a piece-wise affine transformation based on subdomains division has been implemented for modelling the movement of one control rod. In the second approach, a “staircase” strategy has been adopted for simulating the movement of all the three rods featured by the nuclear reactor chosen as case study. The neutron kinetics has been modelled according to the so-called multi-group neutron diffusion, which, in the present case, is a set of ten coupled parametrized parabolic equations (two energy groups for the neutron flux, and eight for the precursors). Both the reduced order models, developed according to the two approaches, provided a very good accuracy compared with high-fidelity results, assumed as “truth” solutions. At the same time, the computational speed-up in the Online phase, with respect to the fine “truth” finite element discretization, achievable by both the proposed approaches is at least of three orders of magnitude, allowing a real-time simulation of the rod movement and control.

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