Homogenization of the linear Boltzmann equation in a domain with a periodic distribution of holes

Consider a linear Boltzmann equation posed on the Euclidian plane with a periodic system of circular holes and for particles moving at speed 1. Assuming that the holes are absorbing -- i.e. that particles falling in a hole remain trapped there forever, we discuss the homogenization limit of that equation in the case where the reciprocal number of holes per unit surface and the length of the circumference of each hole are asymptotically equivalent small quantities. We show that the mass loss rate due to particles falling into the holes is governed by a renewal equation that involves the distribution of free-path lengths for the periodic Lorentz gas. In particular, it is proved that the total mass of the particle system at time t decays exponentially fast as t tends to infinity. This is at variance with the collisionless case discussed in [Caglioti, E., Golse, F., Commun. Math. Phys. 236 (2003), pp. 199--221], where the total mass decays as Const./t as the time variable t tends to infinity.Comment: 29 pages, 1 figure, submitted; figure 1 corrected in new versio

Optimal control in heterogeneous domain decomposition methods

Some new domain decomposition methods (DDM) based on optimal control approach are introduced for the coupling of first- and second-order equations on overlapping subdomains. Several cost functionals and control functions are proposed. Uniqueness and existence results are proved for the coupled problem and the convergence of iterative processes is analyze

Colloidal gelation with variable attraction energy

We present an approximation scheme to the master kinetic equations for aggregation and gelation with thermal breakup in colloidal systems with variable attraction energy. With the cluster fractal dimension $d_{f}$ as the only phenomenological parameter, rich physical behavior is predicted. The viscosity, the gelation time and the cluster size are predicted in closed form analytically as a function of time, initial volume fraction and attraction energy by combining the reversible clustering kinetics with an approximate hydrodynamic model. The fractal dimension $d_{f}$ modulates the time evolution of cluster size, lag time and gelation time and of the viscosity. The gelation transition is strongly nonequilibrium and time-dependent in the unstable region of the state diagram of colloids where the association rate is larger than the dissociation rate. Only upon approaching conditions where the initial association and the dissociation rates are comparable for all species (which is a condition for the detailed balance to be satisfied) aggregation can occur with $d_{f}=3$. In this limit, homogeneous nucleation followed by Lifshitz-Slyozov coarsening is recovered. In this limited region of the state diagram the macroscopic gelation process is likely to be driven by large spontaneous fluctuations associated with spinodal decomposition

Sensitivity of Functionals in Problems of Variational Assimilation of Observational Data

International audienceThe problem of the variational assimilation of observational data is stated for a nonlinear evolutionmodel as a problem of optimal control in order to find the function of initial condition. The operator of themodel, and consequently the optimal solution, depend on parameters that may contain uncertainties. A functional of the solution of the problem of variational data assimilation is considered. Using the method of second-order adjoint equations, the sensitivity of the functional in respect to the model parameters is studied.The gradient of the functional is expressed through solving a “nonstandard” (nonclassical) problem thatinvolves the coupled system of direct and adjoint equations. The solvability of the nonstandard problem usingthe Hessian initial functional of observations is studied. Numerical algorithms for solving the problem andcomputing the gradient of the functional under consideration are developed with respect to the parameters.The results of the studies are applied in the problem of variational data assimilation for a 3D ocean thermodynamic model

Fast simulations of patient-specific haemodynamics of coronary artery bypass grafts based on a POD-Galerkin method and a vascular shape parametrization

In this work a reduced-order computational framework for the study of haemodynamics in three-dimensional patient-specific configurations of coronary artery bypass grafts dealing with a wide range of scenarios is proposed. We combine several efficient algorithms to face at the same time both the geometrical complexity involved in the description of the vascular network and the huge computational cost entailed by time dependent patient-specific flow simulations. Medical imaging procedures allow to reconstruct patient-specific configurations from clinical data. A centerlines-based parametrization is proposed to efficiently handle geometrical variations. POD-Galerkin reduced-order models are employed to cut down large computational costs. This computational framework allows to characterize blood flows for different physical and geometrical variations relevant in the clinical practice, such as stenosis factors and anastomosis variations, in a rapid and reliable way. Several numerical results are discussed, highlighting the computational performance of the proposed framework, as well as its capability to carry out sensitivity analysis studies, so far out of reach. In particular, a reduced-order simulation takes only a few minutes to run, resulting in computational savings of 99% of CPU time with respect to the full-order discretization. Moreover, the error between full-order and reduced-order solutions is also studied, and it is numerically found to be less than 1% for reduced-order solutions obtained with just O(100) online degrees of freedom. (C) 2016 Elsevier Inc. All rights reserved

Combined parameter and model reduction of cardiovascular problems by means of active subspaces and POD-Galerkin methods

In this chapter we introduce a combined parameter and model reduction methodology and present its application to the efficient numerical estimation of a pressure drop in a set of deformed carotids. The aim is to simulate a wide range of possible occlusions after the bifurcation of the carotid. A parametric description of the admissible deformations, based on radial basis functions interpolation, is introduced. Since the parameter space may be very large, the first step in the combined reduction technique is to look for active subspaces in order to reduce the parameter space dimension. Then, we rely on model order reduction methods over the lower dimensional parameter subspace, based on a POD-Galerkin approach, to further reduce the required computational effort and enhance computational efficiency

Parametric free-form shape design with PDE models and reduced basis method

We present a coupling of the reduced basis methods and free-form deformations for shape optimization and design of systems modelled by elliptic PDEs. The free-form deformations give a parameterization of the shape that is independent of the mesh, the initial geometry, and the underlying PDE model. The resulting parametric PDEs are solved by reduced basis methods. An important role in our implementation is played by the recently proposed empirical interpolation method, which allows approximating the non-affinely parameterized deformations with affinely parameterized ones. These ingredients together give rise to an efficient online computational procedure for a repeated evaluation design environment like the one for shape optimization. The proposed approach is demonstrated on an airfoil inverse design problem. © 2010 Elsevier B.V

Engine technical performance influence on car exhaust microparticles grading

Abstract. Present work demonstrates the influence of the engine size, car production year and fuel type on suspended solid particles of the car exhaust. The investigated machine group (N = 21) showed no influence on the particle size distribution caused by engine displacement and type of fuel. It is shown that almost new cars (less than 400 km travelled) are the most frequent source of the particles with an average diameter of about 10 microns or less