Code generation for generally mapped finite elements

Many classical finite elements such as the Argyris and Bell elements have long been absent from high-level PDE software. Building on recent theoretical work, we describe how to implement very general finite-element transformations in FInAT and hence into the Firedrake finite-element system. Numerical results evaluate the new elements, comparing them to existing methods for classical problems. For a second-order model problem, we find that new elements give smooth solutions at a mild increase in cost over standard Lagrange elements. For fourth-order problems, however, the newly enabled methods significantly outperform interior penalty formulations. We also give some advanced use cases, solving the nonlinear Cahn-Hilliard equation and some biharmonic eigenvalue problems (including Chladni plates) using C1 discretizations

Wavelet and Multiscale Methods

Various scientific models demand finer and finer resolutions of relevant features. Paradoxically, increasing computational power serves to even heighten this demand. Namely, the wealth of available data itself becomes a major obstruction. Extracting essential information from complex structures and developing rigorous models to quantify the quality of information leads to tasks that are not tractable by standard numerical techniques. The last decade has seen the emergence of several new computational methodologies to address this situation. Their common features are the nonlinearity of the solution methods as well as the ability of separating solution characteristics living on different length scales. Perhaps the most prominent examples lie in multigrid methods and adaptive grid solvers for partial differential equations. These have substantially advanced the frontiers of computability for certain problem classes in numerical analysis. Other highly visible examples are: regression techniques in nonparametric statistical estimation, the design of universal estimators in the context of mathematical learning theory and machine learning; the investigation of greedy algorithms in complexity theory, compression techniques and encoding in signal and image processing; the solution of global operator equations through the compression of fully populated matrices arising from boundary integral equations with the aid of multipole expansions and hierarchical matrices; attacking problems in high spatial dimensions by sparse grid or hyperbolic wavelet concepts. This workshop proposed to deepen the understanding of the underlying mathematical concepts that drive this new evolution of computation and to promote the exchange of ideas emerging in various disciplines

New lower order mixed finite element methods for linear elasticity

New lower order $H(\textrm{div})$-conforming finite elements for symmetric tensors are constructed in arbitrary dimension. The space of shape functions is defined by enriching the symmetric quadratic polynomial space with the $(d+1)$-order normal-normal face bubble space. The reduced counterpart has only $d(d+1)^2$ degrees of freedom. In two dimensions, basis functions are explicitly given in terms of barycentric coordinates. Lower order conforming finite element elasticity complexes starting from the Bell element, are developed in two dimensions. These finite elements for symmetric tensors are applied to devise robust mixed finite element methods for the linear elasticity problem, which possess the uniform error estimates with respect to the Lam\'{e} coefficient $\lambda$, and superconvergence for the displacement. Numerical results are provided to verify the theoretical convergence rates.Comment: 23 pages, 2 figure

Development of a computational method for inverting dynamic moduli of multilayer systems with applications to flexible pavements

textMost existing computational methods for inverting material properties of multilayer systems have focused primarily on elastic properties of materials or a static approach. Typically, they are based on a two-stage approach: (I) modeling structural responses with a computer program, and (II) estimating layer properties mathematically using the response outputs determined in stage I without interactions with the governing state partial-differential-equation (PDE) of stage I. This two-stage approach may not be accurate and efficient enough for inverting larger scale model parameters. The objective of this research was to develop a computational method to invert dynamic moduli of multilayer systems with applications to flexible pavements under falling weight deflectometer (FWD) tests, thereby advancing existing methods and fostering understanding of material behaviors. This research first developed a finite-element and Newton-Raphson method to invert layer elastic moduli using FWD data. The model improved the moduli seeds estimation and achieved a satisfactory accuracy based on Monte Carlo simulations, addressing the common back-calculation issue of no unique solutions. Consequently, a time-domain finite-element method was developed to simulate dynamic-viscoelastic responses of the multilayer systems under loading pulses. Simulation results demonstrated that the dynamic-viscoelastic-damping-coupled model could emulate structural responses more accurately, thereby advancing existing simulation approaches. By using the dynamic-viscoelastic-response model as one computation module, this research led to the development of a PDE-constrained Lagrangian optimization method to invert dynamic moduli and viscoelastic properties of multilayer systems. The Lagrangian function was used as an objective function, with a regularization term and governing-state PDE constraint. Both the first-order (gradient) and second-order variation (Hessian matrix) of the Lagrangian were computed to satisfy necessary and sufficient optimality conditions, and Armijo rule was modified to determine a stable step length. The developed method improved computation speed significantly, and it is superior for large-scale inverse problems. The model was implemented for evaluating flexible pavements under FWD tests and for inverting the master curve of dynamic moduli of the asphalt layer. Independent computer coding was developed for all numerical methods. The computational methods developed may also be applied to other multilayer systems, such as tissues and sandwich structures at different time and length scales.Civil, Architectural, and Environmental Engineerin

Contributions to the Study of Lithospheric Deformation and Seismicity in Stable Continental Regions

Recently, the field of geophysics has seen increasing recognition of the unique character of deformation and seismicity in stable continental regions (SCRs). However several important questions remain understudied. What controls the locations of earthquakes in SCRs? How well do observations, in SCRs, of elastic strain accumulation and release correlate with each other? How well do they correlate with stresses and geological proxies for rheological variation? The ultimate goal of this study was to better understand stable continental regions like southern Africa, where large earthquakes occur despite not being near plate boundaries, for example the 2017 Mw 6.5 earthquake in Moiyabana, Botswana. One way of studying the stress and strain in stable continental regions is by understanding the surface deformation of the region. This deformation is easily studied using global navigation satellite system (GNSS) velocity data. One of the biggest difficulties when it comes to GNSS data is that it isn't collected on a regular grid, but rather as irregular data points that need to be interpolated. This research investigated multiple interpolation methods and recommended two methods that best replicate the original velocity field (using a well populated dataset from Southeast Asia). These interpolated GNSS data can then be used to determine deviatoric strain in a region, which can in turn be fed into numerical stress models. However, limited GNSS data exist across southern Africa, and therefore topographic data was used to calculate the gravitational potential energy, and in turn the body stress and deviatoric stress for the region. This study also investigated how this deviatoric stress (or deviatoric strain) can be more accurately calculated on a spherical rather than a flat surface, which is particularly important over large study areas. Across southern Africa, data show that deviatoric stress lined up with stress data within mobile belts. This suggests that in these weaker mobile belt crust (such as the Namaqua-Natal and Damara-Chobe belts), gravitational collapse is the dominant driver of deformation, which is in line with conclusions that have been made in previous literature. In other regions, deviatoric stress vectors and stress data do not coincide and therefore there are other forces at play. These observations are obviously restricted by limited data coverage; it remains an open question if areas that have increased deviatoric stress due to gravitational collapse, which are also aligned with the orientation of weak zones, will have elevated strain in the long term