Adaptive Polynomial Local Refinement for Resolving Flame Structure

A new code and methodology are introduced for solving the general relativistic magnetohydrodynamic (GRMHD) equations in fixed background spacetimes using time-explicit, finite-volume discretization. The code has options for solving the GRMHD equations using traditional artificial-viscosity (AV) or non-oscillatory central difference (NOCD) methods, or a new extended AV (eAV) scheme using artificial-viscosity together with a dual energy-flux-conserving formulation. The dual energy approach allows for accurate modeling of highly relativistic flows at boost factors well beyond what has been achieved to date by standard artificial viscosity methods. It provides the benefit of Godunov methods in capturing high Lorentz boosted flows but without complicated Riemann solvers, and the advantages of traditional artificial viscosity methods in their speed and flexibility. Additionally, the GRMHD equations are solved on an unstructured grid that supports local adaptive mesh refinement using a fully threaded oct-tree (in three dimensions) network to traverse the grid hierarchy across levels and immediate neighbors. A number of tests are presented to demonstrate robustness of the numerical algorithms and adaptive mesh framework over a wide spectrum of problems, boosts, and astrophysical applications, including relativistic shock tubes, shock collisions, magnetosonic shocks, Alfven wave propagation, blast waves, magnetized Bondi flow, and the magneto-rotational instability in Kerr black hole spacetimes.

A directional error estimator for adaptive limit analysis

A stencil adaptive algorithm for finite difference solution of incompressible viscous flows

The computational burden associated with multidimensional, multicomponent, numerical solute transport models can be prohibitive. To solve these CPU intensive problems efficiently and accurately, we have implemented and extended a local adaptive grid refinement method of Berger and Oliger (1984) to track error‐prone regions and supply high‐resolution subgrids where they are locally needed while maintaining relatively few nodes elsewhere on a coarse, base grid. Novel features include a unique method for detecting a priori where the numerical error is unacceptable, variable time step control which allows smaller time steps on subgrids than on the base grid, and a modular framework which allows easy exchange of partial differential equation solvers to accommodate different problem formulations. Three examples show how the subgrids track the error‐prone regions, how multiple subgrids are used for geometrically complex front shapes, and how increased resolution is provided for self‐sharpening fronts when multiple reacting species are considered. For all three examples, solutions with accuracies comparable to those achieved with a uniform fine grid are obtained at between 20 and 30% of the computational cost.

S-splines: A simple surface solution for IGA and CAD

Bayesian sparse hierarchical model for image denoising

Fluorescence molecular tomography (FMT) is an imaging modality that exploits the specificity of fluorescent biomarkers to enable 3D visualization of molecular targets and pathways in vivo in small animals. Owing to the high degree of absorption and scattering of light through tissue, the FMT inverse problem is inherently ill-conditioned making image reconstruction highly susceptible to the effects of noise and numerical errors. Appropriate priors or penalties are needed to facilitate reconstruction and to restrict the search space to a specific solution set. Typically, fluorescent probes are locally concentrated within specific areas of interest (e.g., inside tumors). The commonly used L2 norm penalty generates the minimum energy solution, which tends to be spread out in space. Instead, we present here an approach involving a combination of the L1 and total variation norm penalties, the former to suppress spurious background signals and enforce sparsity and the latter to preserve local smoothness and piecewise constancy in the reconstructed images. We have developed a surrogate-based optimization method for minimizing the joint penalties. The method was validated using both simulated and experimental data obtained from a mouse-shaped phantom mimicking tissue optical properties and containing two embedded fluorescent sources. Fluorescence data were collected using a 3D FMT setup that uses an EMCCD camera for image acquisition and a conical mirror for full-surface viewing. A range of performance metrics was utilized to evaluate our simulation results and to compare our method with the L1, L2 and total variation norm penalty-based approaches. The experimental results were assessed using the Dice similarity coefficients computed after co-registration with a CT image of the phantom.

A new adaptive local mesh refinement algorithm and its application on fourth order thin film flow problem

We have developed a differentiable programming framework for truncated hierarchical B-splines (THB-splines), which can be used for several applications in geometry modeling, such as surface fitting and deformable image registration, and can be easily integrated with geometric deep learning frameworks. Differentiable programming is a novel paradigm that enables an algorithm to be differentiated via automatic differentiation, i.e., using automatic differentiation to compute the derivatives of its outputs with respect to its inputs or parameters. Differentiable programming has been used extensively in machine learning for obtaining gradients required in optimization algorithms such as stochastic gradient descent (SGD). While incorporating differentiable programming with traditional functions is straightforward, it is challenging when the functions are complex, such as splines. In this work, we extend the differentiable programming paradigm to THB-splines. THB-splines offer an efficient approach for complex surface fitting by utilizing a hierarchical tensor structure of B-splines, enabling local adaptive refinement. However, this approach brings challenges, such as a larger computational overhead and the non-trivial implementation of automatic differentiation and parallel evaluation algorithms. We use custom kernel functions for GPU acceleration in forward and backward evaluation that are necessary for differentiable programming of THB-splines. Our approach not only improves computational efficiency but also significantly enhances the speed of surface evaluation compared to previous methods. Our differentiable THB-splines framework facilitates faster and more accurate surface modeling with local refinement, with several applications in CAD and isogeometric analysis.

HiFlow3 Version 2.0 continues the path as a multi-purpose finite element software, which provides powerful tools for efficient and accurate solution of a wide range of problems modeled by partial differen- tial equations (PDEs). New features and functionalities, which allow to run numerical simulations with more advanced solution algorithms and discretizations in comparison to previous releases of HiFlow3, have been implemented. These comprise Uncertainty Quantification (UQ), energy-efficient multigrid techniques, Schur complement solvers for saddle-point problems, extended third-party library support, and adaptive local mesh refinement in a parallel computing environment. Furthermore, HiFlow3 has been successfully integrated into advanced and state-of-the-art simulation environments by means of the Medical Simulation Markup Language (MSML), for example.The presented new algorithms and features as well as general under-the-hood improvements have enabled excellent and relevant research activities in the fields of both medical engineering and meteorol- ogy and environmental sciences. The described show cases demonstrate the potential and advantages, which HiFlow3 can offer in performing a numerical simulation by means of finite element methods. Especially, the high performance computing capabilities of HiFlow3 – not only in the mentioned fields of applications, but also in general – have been significantly improved in Version 2.0.

Local adaptive grid refinement is an important technique in finite element methods. Its study can be traced back to the pioneering work [2] in one dimension. In recent years, mathematicians start to prove the convergence and optimal complexity of the adaptive procedure in multi-dimensions. Dorfler [11] first proved an error reduction in the energy norm for the Poisson equation provided the initial mesh is fine enough.

Convergence analysis of sparse LMS algorithms with l1-norm penalty based on white input signal

Mesh refinement for anisotropic diffusion in magnetized plasmas

In the multi‐dimensional numerical simulation of certain multi‐phase fluid flow processes, many phenomena are sufficiently localized and transient that self‐adaptive local grid refinement techniques are necessary to resolve the local physical behaviour. For large‐scale simulation problems, efficiency is the key to the choice of specific adaptive strategies. Several different types of self‐adaptive methods will be discussed and briefly compared. The differences in complexity of the data structure and efficiency of the solution techniques will be explained. Concepts for incorporating these methods in existing large‐scale simulators will also be presented. Man man‐years have been invested in developing large‐scale numerical simulators which do an excellent job of resolving most of the larger length‐scale phenomena in our production processes. Although adaptive local refinement methods are critical for many applications, most techniques which have been developed to date either cannot be incorporated in our codes or are so computationally complex that they destroy the efficiency of existing simulators. Recently, new preconditioning techniques have been developed which allow grid refinement methods to be used efficiently in exisiting codes without disrupting the basic solution process. These techniques are highly amenable to vectorization and parallelization of the algorithms. They are also a first step in allowing complex well models to be incorporated into large field‐scale codes and applications.

In this paper, we develop a fast time-stepping strategy for the Newmark-beta method; the latter is used extensively in structural dynamics. In particular, we speed up the repeated inversion of the linear systems in the Newmark-beta method by implementing and merging four distinct but complementary concepts: (1) voxelization, (2) assembly-free finite element analysis, (3) deflated conjugate gradient, and (4) adaptive local refinement. The resulting assembly-free deflated conjugate gradient (AF-DCG) version of the Newmark-beta is well-suited for large-scale problems, and can be easily ported to multi-core architectures. Numerical experiments demonstrate the efficacy of the proposed method.