Topology optimization in the framework of the linear Boltzmann equation – a method for designing optimal nuclear equipment and particle optics

A Topology Optimization Procedure for Assisting the Design of Nuclear Components

“…It could be the maximization of the radiation dose delivered in an area V of a body for a cancer treatment problem, in which case one would take max Oϕ = ∆V −1 D(E)ϕ(r, E, Ω)drdEdΩ for r ∈ V, with D(E) being the flux-to-dose conversion function and ∆V the volume of area V. The optimization problem min/max Oϕ is subject to some constraints, denoted C(ρ, χ, ϕ(ρ, χ)) = 0, the first one being the requirement that the particle transport obeys the Boltzmann equation, and the other ones being additional constraints, for example, a limitation over the maximum quantity of matter available (weight constraint), or a constraint over the k eff of the system. This formalism is further discussed in Section 1 of [9].…”

“…The derivatives (3) over functions ρ and χ thus become derivatives over parameters ρ i and χ i , whose computation can be performed with the PERT module implemented in the MCNP transport code [10]. Practical instructions on how to use the PERT module for computing derivatives are given in Section 1 of [9] for the computation of ∂L/∂ρ i and in Section P of [9] (within the supplementary material file) for the computation of ∂L/∂χ i . The PERT module relies on the differential operator sampling method, which is faster and more precise than the usual procedure based on the adjoint problem [1,7].…”

“…With this tool, all derivatives (3) can be computed in a single, direct MCNP calculation, which makes the precise resolution of a type (1) problem feasible in a humanly compatible time with the computer resources available in the 2020s. Equation ( 3) is then solved using an iterative algorithm, described in Section 2.2 of [9]. This algorithm starts from a uniform configuration, ρ i = ρ 0 and χ i = χ 0 ∀i, and then improves, iteration after iteration, the design of the structure by incrementing the parameters ρ i and χ i by small, discrete quantities, ±δρ and ±δχ, according to the constraints of Problem (1).…”

“…This algorithm starts from a uniform configuration, ρ i = ρ 0 and χ i = χ 0 ∀i, and then improves, iteration after iteration, the design of the structure by incrementing the parameters ρ i and χ i by small, discrete quantities, ±δρ and ±δχ, according to the constraints of Problem (1). This resolution procedure has been successfully tested over a large range of problems [9]. In these proceedings, we will illustrate its efficiency by solving two problems: the design of a neutron spectrum shaper in Section 3, and the minimization of a critical mass in Section 4.…”

“…At the present times, the most advanced studies are focused on the development of radiation shields for spatial applications and on the fuel organization problem [6][7][8]. In these proceedings, relying on the theoretical framework proposed in [9], we will show that the PERT module implemented in the MCNP code [10][11][12] is precise and fast enough to perform realistic topology optimization of nuclear components. We will illustrate this progress with an application of practical interest, the design of a neutron spectrum shaper, and propose a benchmark on the optimization of the fuel distribution in a critical assembly.…”