Abstract
Reduced basis method has successfully been used in 2D to solve heat transfer parametrized problems. In this work, we present some 3D applications in the same field. We consider two problems, the steady Thermal Fin and the time-dependent Graetz Flow, we compare two reduced-order modelling techniques: Reduced basis and Proper orthogonal decomposition, then we apply a combination of the two strategies in the time-dependent case.
Acknowledgements
We are grateful to Professor. A. Quarteroni (EPFL) and Professor. A.T. Patera (MIT) for support, very useful remarks and suggestions and to the referees who helped in improving this manuscript with their remarks.
Notes
1. A brief historical introduction on RB can be found in [Citation1]. Almroth, Stern and Brogan [Citation2] studied the method for one parameter problem. Then with Noor [Citation3] the method was extended to multi-parameter and non-linear problems in structural analysis and only later in fluid dynamics. Current research is dedicated to the development of a posteriori error estimation procedure and to the development of effective sampling strategies for many parameters spaces in order to improve convergence and computational efficiency [Citation4].
2.We may extend the methodology to the non-compliant case with the introduction of a dual problem [Citation4].
3. The choice of 
and will affect the quality and efficiency of our RB a posteriori error estimators, but will not affect directly our RB output predictions (see [4]).
6. Note that is non-singular under our hypothesis of non-coplanar pre-image nodes and independent of μ; the parametric dependence derives from .
7. The a posteriori error bound must be rigorous (greater or equal to the true error) for all N and all parameters values in the parameter domain . Non-rigorous error indicators may suffice for adaptivity, but not for reliability. Second, the bound must be reasonably sharp. An overly conservative error bound can yield inefficient approximations, typically for N too large. Third, we require efficiency, that is, the cost of the Online evaluation and storage must be independent and should be comparable to the cost of the RB output prediction (see [1]).
8. The is computed by the Successive Constraint Method (SCM) that is an algorithm to construct lower bounds for the coercivity (and in the non-coercive case, inf–sup stability) constant. This method is based on an Offline–Online computational procedure too and it reduces considerably the Online calculation effort. We will not present the method, but the reader can refer to [Citation4,Citation14].
9. A primal-dual formulation is possible to consider non-compliant outputs [Citation16].
10. In this section, we will omit the superscript for the RB approximations.
12. By taking into account the symmetry of the fin configuration, we just consider a half of the fin.
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