Subdivision exterior calculus for geometry processing

The physical models and numerical methodologies of Computational Fluid Dynamics (CFD) are historically linked to the concept of continuous medium and to analysis where continuity, derivation and integration are defined as limits at a point. The first consequence is the need to extend these notions in a multidimensional space by establishing a global inertial frame of reference in order to project the variables there. In recent decades, the emergence of methodologies based on differential geometry or exterior calculus has changed the point of view by starting with the creation of entangled polygonal and polyhedral structures where the variables are located. Mimetic methods and Discrete Exterior Calculus, notably, have intrinsic conservation properties which make them very efficient for solving fluid dynamics equations. The natural extension of this discrete vision relates to the derivation of the equations of mechanics by abandoning the notion of continuous medium. The Galilean frame of reference is replaced by a local frame of reference composed of an oriented segment where the acceleration of the material medium or of a particle is defined. The extension to a higher dimensional space is carried from cause to effect, from one local structure to another. The conservation of acceleration over a segment and the Helmholtz–Hodge decomposition are two essential principles adopted for the derivation of a discrete law of motion. As the fields covered by CFD are increasingly broad, it is natural to return to the deeper meaning of physical phenomena to try a new research or new path which would preserve the properties of current formulations.

This paper discusses computation of time-harmonic wave problems using a mixed formulation and the controllability method introduced by Roland Glowinski. As an example, a scattering problem (in an exterior domain) is considered, and the continuous problem is first formulated in terms of differential forms. Based on the continuous formulation, we write the discrete problem and the controllability algorithm for methods based on both the finite element exterior calculus (FEEC) and the discrete exterior calculus (DEC). As the discrete exterior calculus method provides us with a diagonal “mass matrix”, time-stepping in the DEC approach is remarkably more efficient than in the FEEC approach. For the computations in this paper, we choose the lowest order Whitney elements (a.k.a. Raviart–Thomas elements) for the FEEC approach, and in the DEC discretization we use different diagonal approximations for the Hodge star. Especially, in the DEC approach, a “harmonic Hodge” approximation is used, the derivation of which is based on the time-harmonicity of the problem. Different type of grids are used to study the sensitivity of the solution to the quality of the grid. Putting an effort on meshes regular enough, the computed DEC-solution is as accurate as the FEEC-solution, but reached in the fraction of the time. Both methods seem to be able to keep the solution accuracy rather well in computations with a high wave number (corresponding to a high frequency and a small wave length).

Computational modeling is a key resource to gather insight into physical systems in modern scientific research and engineering. While access to large amount of data has fueled the use of machine learning to recover physical models from experiments and increase the accuracy of physical simulations, purely data-driven models have limited generalization and interpretability. To overcome these limitations, we propose a framework that combines symbolic regression (SR) and discrete exterior calculus (DEC) for the automated discovery of physical models starting from experimental data. Since these models consist of mathematical expressions, they are interpretable and amenable to analysis, and the use of a natural, general-purpose discrete mathematical language for physics favors generalization with limited input data. Importantly, DEC provides building blocks for the discrete analog of field theories, which are beyond the state-of-the-art applications of SR to physical problems. Further, we show that DEC allows to implement a strongly-typed SR procedure that guarantees the mathematical consistency of the recovered models and reduces the search space of symbolic expressions. Finally, we prove the effectiveness of our methodology by re-discovering three models of continuum physics from synthetic experimental data: Poisson equation, the Euler’s elastica and the equations of linear elasticity. Thanks to their general-purpose nature, the methods developed in this paper may be applied to diverse contexts of physical modeling.

We study the optimization of an energy function used by the meshing community to measure and improve mesh quality. This energy is non-traditional because it is dependent on both the primal triangulation and its dual Voronoi (power) diagram. The energy is a measure of the mesh's quality for usage in Discrete Exterior Calculus (DEC), a method for numerically solving PDEs. In DEC, the PDE domain is triangulated and this mesh is used to obtain discrete approximations of the continuous operators in the PDE. The energy of a mesh gives an upper bound on the error of the discrete diagonal approximation of the Hodge star operator. In practice, one begins with an initial mesh and then makes adjustments to produce a mesh of lower energy. However, we have discovered several shortcomings in directly optimizing this energy, e.g. its non-convexity, and we show that the search for an optimized mesh may lead to mesh inversion (malformed triangles). We propose a new energy function to address some of these issues.

This study examines whether the improvement in analyst forecast accuracy around mandatory IFRS adoption is associated with the improvement in the accuracy of financial statement-based forecasts. We find significant out-of-sample improvement in financial statement-based forecast accuracy around mandatory IFRS adoption and significant improvement in analyst forecast accuracy only in countries that made concurrent improvements to financial reporting enforcement. We show that the improvement in analyst forecast accuracy is associated with the improvement in financial statement-based forecast accuracy around IFRS adoption. We also show that analyst forecasts, particularly for firms whose analysts forecast under favorable conditions (i.e., analysts who are less busy with more experience and resources), have a greater association with financial statement-based forecasts, after mandatory IFRS adoption in countries with concurrent changes in enforcement.

Finite element time-domain body-of-revolution Maxwell solver based on discrete exterior calculus

Light scattering by particles large compared to the wavelength of incident light is traditionally solved using ray optics which considers absorption inside the particle approximately, along the ray paths. To study the effects rising from this simplification, we have updated the ray-optics code SIRIS to take into account the propagation of light as inhomogeneous plane waves inside an absorbing particle. We investigate the impact of this correction on traditional ray-optics computations in the example case of light scattering by ice crystals through the extended near-infrared (NIR) wavelength regime. In this spectral range, ice changes from nearly transparent to opaque, and therefore provides an interesting test case with direct connection and applicability to atmospheric remote-sensing measurements at NIR wavelengths. We find that the correction for inhomogeneous waves systematically increases the single-scattering albedo throughout the NIR spectrum for both randomly-oriented, column-like hexagonal crystals and ice crystals shaped like Gaussian random spheres. The largest increase in the single-scattering albedo is 0.042 for hexagonal crystals and 0.044 for Gaussian random spheres, both at λ=2.725 µm. Although the effects on the 4 × 4 scattering-matrix elements are generally small, the largest differences are seen at 2.0 µm and 3.969 µm wavelengths where the correction for inhomogeneous waves affects mostly the backscattering hemisphere of the depolarization-connected P22/P11, P33/P11, and P44/P11. We evaluated the correction for inhomogeneous waves through comparisons against the discrete exterior calculus (DEC) method. We computed scattering by hexagonal ice crystals using the DEC, a traditional ray-optics code (SIRIS3), and a ray-optics code with inhomogeneous waves (SIRIS4). Comparisons of the scattering-matrix elements from SIRIS3 and SIRIS4 against those from the DEC suggest that consideration of the inhomogeneous waves brings the ray-optics solution generally closer to the exact result and, therefore, should be taken into account in scattering by absorbing particles large compared to the wavelength of incident light.

This thesis outlines the construction of several types of structured integrators for incompressible fluids. We first present a vorticity integrator, which is the Hamiltonian counterpart of the existing Lagrangian-based fluid integrator. We next present a model-reduced variational Eulerian integrator for incompressible fluids, which combines the efficiency gains of dimension reduction, the qualitative robustness to coarse spatial and temporal resolutions of geometric integrators, and the simplicity of homogenized boundary conditions on regular grids to deal with arbitrarily-shaped domains with sub-grid accuracy. Both these numerical methods involve approximating the Lie group of volume-preserving diffeomorphisms by a finite-dimensional Lie-group and then restricting the resulting variational principle by means of a non-holonomic constraint. Advantages and limitations of this discretization method will be outlined. It will be seen that these derivation techniques are unable to yield symplectic integrators, but that energy conservation is easily obtained, as is a discretized version of Kelvin's circulation theorem. Finally, we outline the basis of a spectral discrete exterior calculus, which may be a useful element in producing structured numerical methods for fluids in the future.

In this contribution a novel finite element method for the system of first order Maxwell equations in the time domain is introduced. It is based on both a primal and a dual set of finite elements. The resulting discrete curl operators are equivalent to those from basis free methods such as the finite integration method, the discrete exterior calculus, and the Yee-cell method. Using a finite element approach leads to a natural discretisation of the permittivity and permeability operators. Because of the symmetry of the resulting linear system and the particular choice of discretisation, this method, when used in conjunction with a Crank-Nicholson time stepping scheme, conserves the electromagnetic energy exactly.

Tangent vector fields are an essential ingredient in controlling surface appearance for applications ranging from anisotropic shading to texture synthesis and non-photorealistic rendering. To achieve a desired effect one is typically interested in smoothly varying fields that satisfy a sparse set of user-provided constraints. Using tools from Discrete Exterior Calculus, we present a simple and efficient algorithm for designing such fields over arbitrary triangle meshes. By representing the field as scalars over mesh edges (i.e., discrete 1-forms), we obtain an intrinsic, coordinate-free formulation in which field smoothness is enforced through discrete Laplace operators. Unlike previous methods, such a formulation leads to a linear system whose sparsity permits efficient pre-factorization. Constraints are incorporated through weighted least squares and can be updated rapidly enough to enable interactive design, as we demonstrate in the context of anisotropic texture synthesis.

We present an explicit method to compute a generalization of the Fourier Transform on a mesh. It is well known that the eigenfunctions of the Laplace Beltrami operator (Manifold Harmonics) define a function basis allowing for such a transform. However, computing even just a few eigenvectors is out of reach for meshes with more than a few thousand vertices, and storing these eigenvectors is prohibitive for large meshes. To overcome these limitations, we propose a band‐by‐band spectrum computation algorithm and an out‐of‐core implementation that can compute thousands of eigenvectors for meshes with up to a million vertices. We also propose a limited‐memory filtering algorithm, that does not need to store the eigenvectors. Using this latter algorithm, specific frequency bands can be filtered, without needing to compute the entire spectrum. Finally, we demonstrate some applications of our method to interactive convolution geometry filtering. These technical achievements are supported by a solid yet simple theoretic framework based on Discrete Exterior Calculus (DEC). In particular, the issues of symmetry and discretization of the operator are considered with great care.

A class of variational schemes for the hydrodynamic-electrodynamic model of lossless free-electron gas in a quasineutral background is developed for high-quality simulations of surface plasmon polaritons. The Lagrangian density of lossless free-electron gas with a self-consistent electromagnetic field is established, and the dynamical equations with the associated constraints are obtained via a variational principle. Based on discrete exterior calculus, the action functional of this system is discretized and minimized to obtain the discrete dynamics. Newton-Raphson iteration and the biconjugate gradient stabilized method are equipped as a hybrid nonlinear-linear algebraic solver. Instead of discretizing the partial differential equations, the variational schemes have better numerical properties in secular simulations, as they preserve the discrete Lagrangian symplectic structure, gauge symmetry, and general energy-momentum density. Two numerical experiments were performed. The numerical results reproduce characteristic dispersion relations of bulk plasmons and surface plasmon polaritons, and the numerical errors of conserved quantities in all experiments are bounded by a small value after long-term simulations.

Accurate simulation of atmospheric flow in weather and climate prediction models requires the discretization of the governing equations to have a number of desirable properties. Although these properties can be achieved relatively straightforwardly on a latitude-longitude grid, they are much more challenging on the various quasi-uniform spherical grids that are now under consideration. A recently developed scheme—called TRiSK—has these desirable properties on grids that have an orthogonal dual. The present work extends the TRiSK scheme into a more general framework suitable for grids that have a nonorthogonal dual, such as the equiangular cubed sphere. We also show that this framework fits within the wider framework of mimetic discretizations and discrete exterior calculus. One key ingredient is the definition of certain mapping operators that are discrete analogues of the Hodge star operator, enabling the definition of a compatible inner product. Discrete Coriolis terms are also included within the mimet...

Introduction: MRCP has achieved widespread success as a diagnostic tool for biliopancreatic cancer. Aim of the study: is to assess the diagnostic quality and overall accuracy of MRCP techniques for the evaluation of biliopancreatic cancer. Patients and Methods: from 266 patients (115 male, 151 female; mean age 50), 63 suspected for biliary tree or pancreatic diseases cancer and histopathologically confirmed, were enrolled in this study. MRCP protocols applied for imaging of biliary and pancreatic tree were; T2-weighted Fast spin echo (FSE) sequence on the axial plane, three-dimensional, fatsuppressed, heavily T2-weighted FSE sequence with multislab acquisition mode, two-dimensional thick single slab projectional images, three-dimensional reconstruction algorithms. Results: among the patients, jaundice and abdominal pain were the commonest clinical complaints. Three-dimensional, fat-suppressed, heavily T2-weighted FSE sequence with multislab acquisition mode shows a significant (p<0.05) improvement in accuracy to from 88.1% to 92.9% compared to two-dimensional thick single slab projectional images. Moreover, there is a significant increase in sensitivity from 90.5% to 95.2% was seen; specificity, too, increased from 85.7% to 90.5%. Furthermore, Overall bile duct visualization with the 3D fat-suppressed, heavily T2 FSE with multislab acquisition technique was rated significantly higher (mean score = 4.7) versus 2D thick single slab projectional images (mean score= 4) (P < 0.05). Conclusion: MRCP plays a critical role in the differential diagnosis with significant improvement in the accuracy of the diagnosis of biliopancreatic cancer.

To retrospectively assess the utility of fusion of T2-weighted images (T2WI) and high b-value diffusion-weighted images (DWI) for prostate cancer detection and localization. In this IRB-approved HIPAA-compliant study, 42 patients with prostate cancer underwent MRI including multiplanar T2WI and axial DWI before prostatectomy. Two independent radiologists first assessed multiplanar T2WI and axial DWI(b-1000) images and recorded whether tumor was present in each sextant. Axial T2WI was then fused with axial DWI(b-1000) images, and the radiologists re-evaluated each sextant for tumor. Accuracy was compared using generalized estimating equations based on a binary logistic regression model. The accuracy, sensitivity, specificity, PPV, and NPV for tumor detection on a sextant-basis using separate and fused image sets was 65.1%, 50.8%, 78.0%, 67.8%, and 63.6% and 71.0%, 60.8%, 80.3%, 73.7%, and 69.3%, respectively, for reader 1, and 54.0%, 42.5%, 64.4%, 52.0%, and 55.2%, and 61.1%, 56.7%, 65.2%, 59.6%, and 62.3%, respectively, for reader 2. The improvements in accuracy, sensitivity, and NPV using fused images were statistically significant for both readers, as was the improvement in PPV for reader 2 (P ranging from <0.0001 to 0.041). With either separate or fused images, there was greater sensitivity for tumors of higher grade or larger size (P ranging from <0.001 to 0.099). Fusion of T2WI and high b-value DWI resulted in significant improvements in sensitivity and accuracy for tumor detection on a sextant-basis, with similar specificity.