Acoustic shape optimization using energy stable curvilinear finite differences

A gradient-based method for shape optimization problems constrained by the acoustic wave equation is presented. The method makes use of high-order accurate finite differences with summation-by-parts properties on multiblock curvilinear grids to discretize in space. Representing the design domain through a coordinate mapping from a reference domain, the design shape is obtained by inversion of the discretized coordinate map. The adjoint state framework is employed to efficiently compute the gradient of the loss functional. Using the summation-by-parts properties of the finite difference discretization, we prove stability and dual consistency for the semi-discrete forward and adjoint problems. Numerical experiments verify the accuracy of the finite difference scheme and demonstrate the capabilities of the shape optimization method on two model problems with real-world relevance

Adjoint-based inversion for stress and frictional parameters in earthquake modeling

We present an adjoint-based optimization method to invert for stress and frictional parameters used in earthquake modeling. The forward problem is linear elastodynamics with nonlinear rate-and-state frictional faults. The misfit functional quantifies the difference between simulated and measured particle displacements or velocities at receiver locations. The misfit may include windowing or filtering operators. We derive the corresponding adjoint problem, which is linear elasticity with linearized rate-and-state friction with time-dependent coefficients derived from the forward solution. The gradient of the misfit is efficiently computed by convolving forward and adjoint variables on the fault. The method thus extends the framework of full-waveform inversion to include frictional faults with rate-and-state friction. In addition, we present a space-time dual-consistent discretization of a dynamic rupture problem with a rough fault in antiplane shear, using high-order accurate summation-by-parts finite differences in combination with explicit Runge--Kutta time integration. The dual consistency of the discretization ensures that the discrete adjoint-based gradient is the exact gradient of the discrete misfit functional as well as a consistent approximation of the continuous gradient. Our theoretical results are corroborated by inversions with synthetic data. We anticipate that adjoint-based inversion of seismic and/or geodetic data will be a powerful tool for studying earthquake source processes; it can also be used to interpret laboratory friction experiments.Comment: Updated title, added additional references, provided additional details in sections 1 and 5, fixed typo

Summation-by-Parts Finite Difference Methods for Wave Propagation and Earthquake Modeling

Waves manifest in many areas of physics, ranging from large-scale seismic waves in geophysics down to particle descriptions in quantum physics. Wave propagation may often be described mathematically by partial differential equations (PDE). Unfortunately, analytical solutions to PDEs are in many cases notoriously difficult to obtain. For this reason, one turns to approximate solutions obtained through numerical methods implemented as computer algorithms. In order for a numerical method to be useful in predictive simulations, it should be stable and accurate. Stability of the method ensures that small errors in the approximation do not grow exponentially. Accuracy together with stability ensures that increased resolution in the simulation results in decreased error in the approximation. The numerical methods considered in this thesis are finite difference methods satisfying a summation-by-parts (SBP) property. Finite difference methods are well suited for wave propagation problems in that they provide high accuracy at low computational cost. The SBP property additionally facilitates the construction of provably stable high-order accurate approximations. This thesis continues the development of SBP finite difference methods for wave propagation problems. Paper I presents a finite difference method for modeling induced seismicity, i.e., earthquakes caused by human activity. Paper II develops a high-order accurate finite difference method for shock waves modeled by scalar conservation laws. In Paper III, new SBP finite difference operators with increased accuracy and efficiency for surface and interface waves are derived. In Papers IV - V numerical methods for inverse problems governed by wave equations are considered, where unknown model parameters are reconstructed by fitting the numerical solution to known data. Specifically, Paper IV presents a method for acoustic shape optimization, while Paper V presents an inversion method for frictional parameters used in earthquake modeling

Numerical simulation of the Dynamic Beam Equation using the SBP-SAT method

A stable boundary treatment of the dynamic beam equation (DBE) with two different sets of boundary conditions has been conducted using the summation-by-parts-simultaneous-approximation-term (SBP-SAT) method. As the DBE involves a fourth derivative in space the numerical boundary treatment is highly non-trivial. Using SBP-SAT operators together with suitable time integration schemes the DBE has been simulated and a convergence study has been made. The results show that the SBP-SAT method produces a stable discretistation that is accurate enough to capture the dispersive nature of the dynamic beam equation. In additions simulations were made presenting the importance of a stable boundary treatment showing that the numerical solutions diverge when the boundaries were not handled correctly

Numerical simulation of the Dynamic Beam Equation using the SBP-SAT method

A stable boundary treatment of the dynamic beam equation (DBE) with two different sets of boundary conditions has been conducted using the summation-by-parts-simultaneous-approximation-term (SBP-SAT) method. As the DBE involves a fourth derivative in space the numerical boundary treatment is highly non-trivial. Using SBP-SAT operators together with suitable time integration schemes the DBE has been simulated and a convergence study has been made. The results show that the SBP-SAT method produces a stable discretistation that is accurate enough to capture the dispersive nature of the dynamic beam equation. In additions simulations were made presenting the importance of a stable boundary treatment showing that the numerical solutions diverge when the boundaries were not handled correctly

Numerical simulation of the Dynamic Beam Equation using the SBP-SAT method

A stable boundary treatment of the dynamic beam equation (DBE) with two different sets of boundary conditions has been conducted using the summation-by-parts-simultaneous-approximation-term (SBP-SAT) method. As the DBE involves a fourth derivative in space the numerical boundary treatment is highly non-trivial. Using SBP-SAT operators together with suitable time integration schemes the DBE has been simulated and a convergence study has been made. The results show that the SBP-SAT method produces a stable discretistation that is accurate enough to capture the dispersive nature of the dynamic beam equation. In additions simulations were made presenting the importance of a stable boundary treatment showing that the numerical solutions diverge when the boundaries were not handled correctly

Summation-by-Parts Finite Difference Methods for Wave Propagation and Earthquake Modeling

Waves manifest in many areas of physics, ranging from large-scale seismic waves in geophysics down to particle descriptions in quantum physics. Wave propagation may often be described mathematically by partial differential equations (PDE). Unfortunately, analytical solutions to PDEs are in many cases notoriously difficult to obtain. For this reason, one turns to approximate solutions obtained through numerical methods implemented as computer algorithms. In order for a numerical method to be useful in predictive simulations, it should be stable and accurate. Stability of the method ensures that small errors in the approximation do not grow exponentially. Accuracy together with stability ensures that increased resolution in the simulation results in decreased error in the approximation. The numerical methods considered in this thesis are finite difference methods satisfying a summation-by-parts (SBP) property. Finite difference methods are well suited for wave propagation problems in that they provide high accuracy at low computational cost. The SBP property additionally facilitates the construction of provably stable high-order accurate approximations. This thesis continues the development of SBP finite difference methods for wave propagation problems. Paper I presents a finite difference method for modeling induced seismicity, i.e., earthquakes caused by human activity. Paper II develops a high-order accurate finite difference method for shock waves modeled by scalar conservation laws. In Paper III, new SBP finite difference operators with increased accuracy and efficiency for surface and interface waves are derived. In Papers IV - V numerical methods for inverse problems governed by wave equations are considered, where unknown model parameters are reconstructed by fitting the numerical solution to known data. Specifically, Paper IV presents a method for acoustic shape optimization, while Paper V presents an inversion method for frictional parameters used in earthquake modeling

A High Order Finite Difference Method for Simulating Earthquake Sequences in a Poroelastic Medium

Induced seismicity (earthquakes caused by injection or extraction of fluids in Earth's subsurface) is a major, new hazard in the United States, the Netherlands, and other countries, with vast economic consequences if not properly managed. Addressing this problem requires development of predictive simulations of how fluid-saturated solids containing frictional faults respond to fluid injection/extraction. Here we present a numerical method for linear poroelasticity with rate-and-state friction faults. A numerical method for approximating the fully coupled linear poroelastic equations is derived using the summation-by-parts-simultaneous-approximation-term (SBP-SAT) framework. Well-posedness is shown for a set of physical boundary conditions in 1D and in 2D. The SBP-SAT technique is used to discretize the governing equations and show semi-discrete stability and the correctness of the implementation is verified by rigorous convergence tests using the method of manufactured solutions, which shows that the expected convergence rates are obtained for a problem with spatially variable material parameters. Mandel's problem and a line source problem are studied, where simulation results and convergence studies show satisfactory numerical properties. Furthermore, two problem setups involving fault dynamics and slip on faults triggered by fluid injection are studied, where the simulation results show that fluid injection can trigger earthquakes, having implications for induced seismicity. In addition, the results show that the scheme used for solving the fully coupled problem, captures dynamics that would not be seen in an uncoupled model. Future improvements involve imposing Dirichlet boundary conditions using a different technique, extending the scheme to handle curvilinear coordinates and three spatial dimensions, as well as improving the high-performance code and extending the study of the fault dynamics.