Tensorized Discontinuous Isogeometric Analysis Method for the 2-D Time-Independent Linearized Boltzmann Transport Equation

TL;DR

This paper introduces a tensorized discontinuous isogeometric analysis method for 2D linearized Boltzmann transport equations, achieving significant data compression and enabling high-fidelity simulations despite increased computational time.

Contribution

The novel TDIGA method applies tensor train format to efficiently solve high-dimensional neutron transport problems with improved data compression and solution accuracy.

Findings

Tensor train format compresses operators from petabytes to megabytes.

Mixed formats mitigate time-to-solution issues for boundary operators.

Method enables high-fidelity transport simulations on complex IGA meshes.

Abstract

We present the novel Tensorized Discontinuous Isogeometric Analysis (TDIGA) method applied to the discontinuous Galerkin (DG) time-independent 2-D linearized Boltzmann transport equation (LBTE) with higher-order scattering, discretized with discrete ordinates in angle, multigroup in energy, and isogeometric analysis (IGA) in space. We formulate operator assembly in the tensor train (TT) format, producing seven-dimensional operators for both fixed-source and $k$-eigenvalue neutron transport problems solved using the restarted Generalized Minimum Residual Method (GMRES) and power iteration with an uncompressed solution vector. Our results on single-patch homogeneous and multi-patch heterogeneous problems, including a cruciform-shaped fuel array inspired by advanced reactor fuel designs, demonstrate the TT format's ability to compress interior operators from petabytes to megabytes, whereas the Compressed Sparse Row (CSR) matrix format requires gigabytes of storage. However, highly coupled boundary operators present a significant challenge for TT. Despite the storage savings, TT formatted operators increase time-to-solution relative to CSR as an uncompressed solution vector forces operator-vector product scaling of $O(dr^2N^d\log(N))$ for TT while CSR scales at $O(\text{nnz})$. We mitigate this discrepancy by using mixed formats with interior operators in TT, while high-rank boundary operators remain in CSR format. We compare all results to Monte Carlo (MC) and analytic reference solutions. While CSR remains $<10\times$ faster than this mixed format, the TDIGA method enables high-fidelity transport for expensive high-order IGA meshes.