Reconstruction of Lava Rheology in a Thin-Layer Model of Viscous Flow

Abstract

we consider the problem of estimating the rheological properties of a thin layer of viscous incompressible fluid flowing over a prescribed surface. The problem is formulated as an inverse problem for a model in which the fluid viscosity depends on spatial coordinates. We assume that the problem is ill-posed, requiring specialized numerical methods to ensure solution stability.
We propose a variational approach, replacing the original problem with an extremal problem that minimizes a functional representing the mismatch between observed parameters and the corresponding model solution. The solution is approximated sequentially through a series of initial control problems, formulated as nonlinear systems of partial differential equations with fully defined parameters. To minimize the mismatch functional, we apply a linearized conjugate gradient method in the Polak-Ribiere implementation. The gradient and descent step are computed analytically, significantly reducing computational cost.
We integrate the partial differential equation systems using the finite volume method for domains of various geometries. The numerical simulation algorithms are verified using the OpenFOAM computing package. The resulting computer codes are optimized for execution on computing clusters with both shared and distributed memory on Linux-based CPUs.