Rubén Panta Pazos

Convergence in transport theory

Abstract In this work we prove the convergence of the LTSN method for the steady-state and time-dependent one-dimensional linear transport equation employing the C 0 semigroup theory and discrete schemes approach.

Convergence of the LTSN method: Approach of C0 semigroups

Abstract In this work we report the convergence of the LTS N solution for the steady state using the C 0 semigroup approach first considering that the total cross section and scattering kernel are both independent with respect to the spatial variable. Then we consider that the total cross section and scattering kernel hinge on the spatial variable and apply the procedure of Pazy and Tanabe [Pazy 1983] and [Tanabe 1979].

Solutions to the multidimensional linear transport equation by the spectral method

The spectral method is used to develop a solution for multidimensional transport problems for neutral particles in cartesian geometry. The procedure is based on the expansion of the angular flux in a truncated series of orthogonal polynomials that results in the transformation of the multidimensional problem into a set of one-dimensional problems, whose solutions are well established. The convergence of this approach is studied in the context of the multidimensional discrete-ordinates equations.

Qualitative analysis of the SN approximations of the transport equation and combined conduction-radiation heat transfer problem in a slab

Abstract A review is presented on the recent work carried out by our research group on mathematical aspects of the existence and uniqueness of solutions of the one-dimensional steady-state transport equation as well as an analytical study of the nonlinear radiative transfer equation in a slab. Dependence on control parameters about the solutions of the one-dimensional S N , approximations of the transport equation is also addressed.

On the convergence of the spherical harmonics approximations

Abstract The convergence of the spherical harmonics approximations using the isomorphism between two functional spaces and the approximation theorem of the C0-semigroup theory is proved.

Determination of a closed-form solution for the multidimensional transport equation using a fractional derivative

Abstract In this work we construct a closed-form solution for the multidimensional transport equation rewriten in integral form which is expressed in terms of a fractional derivative of the angular flux. We determine the unknown order of the fractional derivative comparing the kernel of the integral equation with the one of the Riemann–Liouville definition of fractional derivative. We report numerical simulations.

Behavior of a Sequence of Geometric Transformations for a Truncated Ellipsoid Geometry in Transport Theory

The neutron transport equation has been studied from different approaches, in order to solve different situations. The number of methods and computational techniques has increased recently. In this work we present the behavior of a sequence of geometric transformations evolving different transport problems in order to obtain solve a transport problem in a truncated ellipsoid geometry and subject to known boundary conditions. This scheme was depicted in 8 , but now is solved for the different steps. First, it is considered a rectangle domain that consists of three regions, source, void and shield regions 5 . Horseshoe domain: for that it is used the complex function: f:D→C,defined as f(z)=12ez+1ezwhereD=z∊C−0.5≤Re(z)≤0.5,−12π≤Im(z)≤12π (0.1) The geometry obtained is such that the source is at the focus of an ellipse, and the target coincides with the other focus. The boundary conditions are reflective in the left boundary and vacuum in the right boundary. Indeed, if the eccentricity is a number between 0,95 and 0,99, the distance between the source and the target ranges from 20 to 100 length units. The rotation around the symmetry axis of the horseshoe domain generates a truncated ellipsoid, such that a focus coincides with the source. In this work it is analyzed the flux in each step, giving numerical results obtained in a computer algebraic system. Applications: in nuclear medicine and others.Copyright

Treatment of Noise in Experimental Transport Measurements Plots With Discrete Wavelet Transforms

In this work it is applied the wavelet transform method [2] in order to reduce diverse type of noises of experimental measurement plots in transport theory. First, suppose that a problem is governed by the transport equation for neutral particles, and an unknown perturbation occurs. In this case, the perturbation can be associated to the source, or even to the flux inside the domain X. How is the behavior of the perturbed flux in relation to the flux without the perturbation? For that, we employ the wavelet transform method in order to compress the angular flux considered as a 1D, or n-th dimensional signal ψ. The compression of this signal can be performed up to some a convenient order (that depends of the length of the signal). Now, the transport signal is decomposed as [9, 11]: ψ=〈am|dm|dm−1|dm−2|⋯|d2|d1〉 where ak represents the sub signal of k-th level generated by the low-pass filter associated to the discrete wavelet transform (DWT) chosen, and dk the sub signal of k-th level generated by the high-pass filter associated to the same DWT. It is applied basically the Haar, Daub4 and Coiflet wavelets transforms. Indeed, the sub signal am cumulates the energy, for this work of order 96% of the original signal ψ. A thresholding algorithm provides treatment for the noise, with significant reduction in the compressed signal. Then, it is established a comparison with a base of data in order to identify the perturbed signal. After the identification, it is recomposed the signal applying the inverse DWT. Many assumptions can be established: the rate signal-to-noise is properly high, the base of data must contain so many perturbed signals all with the same level of compression. The problem considered is for perturbations in the signal. For measurements the problem is similar, but in this case the unknown perturbations are generated by the apparatus of measurements, problems in experimental techniques, or simply by random noises. With the same above assumptions, the DWT is applied. For the identification, it is used a method evolving statistical and metric techniques. It is given some results obtained with an algebraic computer system.Copyright

Hybrid Methods Approach for Solving Problems in Transport Theory

In this work the hybrid methods approach is introduced in order to solve some problems in Transport Theory for different geometries. The transport equation is written as: ∂ψ∂t(x,v,t)+v·∇ψ(x,v,t) +h(x,μ)ψ(x,v,t)= = V k(x,v,v′)ψ(x,v′,t)dv′ +q(x,v,t), in ΩTψ(x,v,0) = φ0(x,v), in ∂Ω×Vψ(x,v,t) = φ(x,v,t), in ∂Ω×V×R (1) where x represents the spatial variable in a domain D, v an element of a compact set V, ψ is the angular flux, h(x, v) the collision frequency, k(x, v, v’) the scattering kernel function and q(x, v) the source function. If ψ does not depend on the time, it is said that the problem (1) is a steady transport problem. Once the problem is defined, including the boundary conditions, it is disposed a set of chained methods in order to solve the problem. Between the different alternatives, an optimal scheme for the resolution is chosen. Two illustrations are given. For two-dimensional geometries it is employed a hybrid analytical and numerical method, for transport problems: conformal mapping first, then the solution in a proper geometry (rectangular for example). Each of the following two techniques is then applied, Krylov subspaces method or spectral-LTSN method. For three-dimensional problems also it is used a hybrid analytical and numerical method, for problems with more complex geometries: a homotopy between the original boundaries (piecewise surfaces) and another (a parallelepiped for example). Then each of two techniques are applied, Krylov subspaces method or nodal-LTSN method. In this case, the design of new geometries for reactors is a straightforward task. En each case, the domain consist of three regions, one of the source, other is the void region and the third one is a shield domain. The results are obtained both with an algebraic computer system and with a language of high level. An important extension is the study and treatment of transport problems for domains with irregular geometries, between them Lipschitzian domains. One remarkable fact of this work is the combination of different modeling and resolution techniques to solve some transport problems.Copyright

Finding the Minimun of the Quadratic Functional in Variational Approach in Transport Theory Problems

In this work it is reviewed the variational approach for some Transport Problems. Let X be a convex domain in Rn , and V a compact set. For that, it is considered the following equation: ∂ψ∂t(x,v,t)+v·∇ψ(x,v,t)+h(x,μ) ψ(x,v,t)== Vk(x,v,v′)ψ(x,v′,t)dv′ +q(x,v,t) (1) where x represents the spatial variable in a domain D, v an element of a compact set V, Ψ is the angular flux, h(x,v) the collision frequency, k(x,v,v’) the scattering kernel function and q(x,v) the source function. It is put the attention in the construction of the quadratic functional J which appears in variational approaches for transport theory (for example, the Vladimirov functional). Some properties of this functional in a proper functional framework, in order to determine the minimum for J are considered. First, the general formulation is studied. Then an algorithm is given for minimizing the functional J for two remarkable problems: spherical harmonic method and spectral collocation method. A program associated to this algorithm is worked in a computer algebraic system, and also was depeloped a version in a high level language.Copyright