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 Preprint submitted to Comput. Meth. Appl. Mech. Eng. February 4, 2012 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.

The aim of this work is to solve parametrized partial differential equations in computational domains represented by networks of repetitive geometries by combining reduced basis and domain decomposition techniques. The main idea behind this approach is to compute once, locally and for few reference shapes, some representative finite element solutions for different values of the parameters and with a set of different suitable boundary conditions on the boundaries: these functions will represent the basis of a reduced space where the global solution is sought for. The continuity of the latter is assured by a classical domain decomposition approach. Test results on Poisson problem show the flexibility of the proposed method in which accuracy and computational time may be tuned by varying the number of reduced basis functions employed, or the set of boundary conditions used for defining locally the basis functions. The proposed approach simplifies the pre-computation of the reduced basis space by splitting the global problem into smaller local subproblems. Thanks to this feature, it allows dealing with arbitrarily complex network and features more flexibility than a classical global reduced basis approximation where the topology of the geometry is fixed

In this paper the reduced basis method is utilized to solve multiobjective optimization problems governed by linear variational equations. These problems often arise in practical applications, where the quality of the system behavior has to be measured by more than one criterium. For the numerical solution the weighting sum method is applied. This approach leads to an algorithm, where many parameterized quadratic optimization problems are solved very efficiently by a appropriate reduced basis approximation. Further, the number of parameter variations is reduced by a sensitivity analysis for the parameterized objective.

Abstract. In this paper a reduced-order strategy is applied to solve a multiobjective optimal control problem governed by semilinear parabolic partial differential equations. These problems often arise in practical applications, where the quality of the system behavior has to be measured by more than one criterium. The weighted sum method is exploited for defining scalar-valued nonlinear optimal control problems built by introducing additional optimization parameters. The optimal controls corresponding to specific choices of the optimization parameters are efficiently computed by the reduced-order method. The accuracy is guaranteed by an a-posteriori error estimate.

Summary Automated managed-pressure drilling (MPD) is a method to enhance downhole pressure-control performance and safety during drilling operations. It is becoming more common to use model-based simulation for the evaluation of pressure-control systems designed for MPD automation before using those in the field. This demands a representative hydraulics-simulation model that captures the relevant aspects of a drilling system. This paper presents such a model and an approach to numerically implement that model for simulation studies. The complexity of this simulation model should be limited, first, to support effective numerical implementation and, second and most importantly, to allow for the analysis of the behavior and performance of the automated pressure-control systems during the controller-design phase. To this end, aspects of a drilling system that can considerably affect the performance of the automated MPD system are captured in the model. This hydraulics model incorporates both the distributed and multiphase-flow nature of a drilling system. Moreover, it captures nonlinear boundary conditions at the inlet of the drillstring, at the drill bit, and choke manifold, and also the variations in the cross-sectional area of the flow path. Model validations against field data from real-life MPD operations and simulations of industry-relevant scenarios indicate that these aspects are effectively captured in the model and preserved during the numerical implementation.