ABSTRACT. In this study we demonstrate the applicability of Firedrake for geodynamical problems, with a focus on mantle and lithosphere dynamics. The accuracy and efficiency of the approach is confirmed via comparisons against a suite of analytical and benchmark cases, of systematically increasing complexity. In addition, the ease of application to different physical (e.g. complex non-linear rheologies, compressibility) and geometrical (2-D and 3-D Cartesian and spherical domains) scenarios is highlighted. Furthermore, good weak parallel scalability is demonstrated on up to 12288 compute cores. Finally, preliminary results from a simulation of global mantle convection at realistic convective vigour that incorporates 200 Myr of plate motion history as a kinematic surface boundary condition, highlights its suitability for addressing frontier questions in global mantle dynamics research.

ABSTRACT. In this presentation we show a computational-efficient Firedrake-based software to perform time-domain full waveform inversion (FWI). FWI is an inverse approach commonly used in geophysical exploration studies to investigate Earth’s physical properties. The FWI approach requires several iterations of costly acoustic or elastic wave simulations in domains representing the subsurface of the Earth. Here, a novel approach is presented to reduce the computational cost of FWI runs by using unstructured triangular meshes and higher-order mass lumped elements. We rely on automatic mesh refinement that adapts the triangular elements to subsurface discontinuities and higher-order basis functions that allows us to reduce the number of degrees-of-freedom necessary to perform time-domain FWI and becomes feasible for 3D. We assess the performance of our approach by running synthetic FWI benchmarks. We finish the presentation highlighting ongoing work with elastic FWI.

This research was carried out in association with the ongoing R&D project registered as ANP 20714-2, “Software technologies for modeling and inversion, with applications in seismic imaging” (University of São Paulo / Shell Brasil / ANP).

ABSTRACT. In seismic imaging, Full-Waveform Inversion (FWI) has the goal to minimize the difference between the real wave and synthetic data storage in the receivers. This difference is referred to as the misfit function, in which its gradient with respect to the velocity model is used in the optimization process. That is performed using the augmented Lagrangian functional, where a mathematical expression of the misfit gradient depends on the Lagrange multiplier that is the solution of the adjoint wave equation. The adjoint wave equation may be obtained in continuous space, in discrete space, or using automatic differentiation. In this context, this work proposes to present two-dimensional seismic imaging on solving the FWI with automatic differentiation. We intend to compare the convergence of the optimization process with the case in which the discrete adjoint wave equation is used, firstly by adopting a simpler velocity model. Next, for a more realistic velocity model well-known in the literature: the Marmousi model. To do that, a software development that is implemented on top of Firedrake is used, where the continuous Galerkin finite elements are adopted. Besides that, triangular elements are used in spatial discretization, a fully explicit, variable higher-order mass lumping method is used to solve the forward wave equation.

ABSTRACT. The Morse-Ingard equations model the pressure and temperature of an excited gas, an essential part of a larger model of trace gas sensors. They are a coupled pair of time-harmonic Helmholtz-like equations in the exterior of the sensor geometry. As such, they face many of the same issues as Helmholtz equations -- in particular, one must impose a suitable artificial boundary condition to truncate the computational domain.

Following recent work in [1] for the Helmholtz operator, we pose an exact boundary condition on the external boundary. This avoids many of the complexities and/or inaccuracies of artificial boundary conditions, but it does require the evaluation of layer potentials, which involve boundary convolution with the Green's functions for the Helmholtz operator at certain complex frequencies. The resulting operators can be evaluated efficiently in a matrix-free fashion, using a fast multipole method for the convolutions, and preconditioned with a suitable solver for the local part of the operator.

This application provides a nontrivial use case for Firedrake external operators, and we will describe the current work in progress to evaluate layer potentials through the UFL interface. In addition, the problem concurrently utilizes several other interesting Firedrake features such as complex arithmetic, curvilinear geometry, matrix-free operators.

[1] Kirby, Robert C., Andreas Klöckner, and Ben Sepanski. "Finite elements for Helmholtz equations with a nonlocal boundary condition." SIAM Journal on Scientific Computing 43.3 (2021): A1671-A1691.

ABSTRACT. A finite element method for simulating the interaction of a planar non-shearable rod immersed in generalized Newtonian fluids is proposed. The method is suitable for simulating microscopic devices, elongated organisms or the appendices (flagella, cilia) of more complex ones in the soft-bio-matter realm, which are subject to very large displacements and deformations as they interact with the surrounding fluid. The method does not rely on the regularizing effect of inertia of either the solid or the fluid. A mechanism for active response is also incorporated by allowing the strain energy of the solid to explicitly depend on time. For discretization of the rod’s problem, one-dimensional Hermite elements are adopted. As for the fluid problem, a stabilized equal order formulation is implemented. The interaction between the structure and the fluid is solved by suitably constructing the space of kinematically admissible motions. The fluid mesh is boundary-fitted, with remeshing at each time step. Several very challenging examples are shown to illustrate the accuracy and robustness of the method. Also, several details of the Firedrake implementation are provided.

ABSTRACT. We seek to investigate the effectiveness of different finite element methods for solving the biharmonic problem. To do so I have used Firedrake to implement a range of methods and determine their accuracy and computational complexity while varying the fineness of the mesh and the polynomial degree. To achieve this I implemented the different methods in classes, all children of the same parent class, in order to share all functionality that is common between the methods. In the future we intend to apply this to work as an inner product for optimisation problems.