Meshes optimized for discrete exterior calculus (DEC).

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.

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).

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.

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 introduces a new computational method to solve differential equations on subdivision surfaces. Our approach adapts the numerical framework of Discrete Exterior Calculus (DEC) from the polygonal to the subdivision setting by exploiting the refin-ability of subdivision basis functions. The resulting Subdivision Exterior Calculus (SEC) provides significant improvements in accuracy compared to existing polygonal techniques, while offering exact finite-dimensional analogs of continuum structural identities such as Stokes' theorem and Helmholtz-Hodge decomposition. We demonstrate the versatility and efficiency of SEC on common geometry processing tasks including parameterization, geodesic distance computation, and vector field design.

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...

Freeform surface models are conventionally used to model sheet metal components, such as automobile body parts. Finite element meshes generated automatically for such models have poor quality around small detailed features. Manual correction of the mesh is extremely tedious. An approach presently receiving attention in industry aims to alleviate this problem by automatically simplifying these features in the surface model such that an acceptable mesh is automatically generated. Simplification involves recognition of the feature and modification of its geometry or complete suppression of the feature. Since features such as holes, notches, etc. are punched after the basic shape has been formed, such a simplification will also help in the modelling of forming dies and molds. The ability to detect features will also allow part comparison and classification in surface models. This paper proposes techniques to directly query the CAD data structure to recognize and suppress two basic features, viz. holes and fillets in freeform surface models. It further demonstrates how these techniques can be extended to suppress compound features that are composed of a combination of basic features. Results of a software implementation for the same are discussed with suitable examples and the improvement in mesh quality is demonstrated.

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.

We revisit the theory of Discrete Exterior Calculus (DEC) in 2D for general triangulations, relying only on Vector Calculus and Matrix Algebra. We present DEC numerical solutions of the Poisson equation and compare them against those found using the Finite Element Method with linear elements (FEML).

Truncation error expressions for 1D and 2D Burgers’ equations are derived in generalized coordinates. Examination of these truncation error expressions provides insight into the effects of mesh quality (mesh stretching in 1D and mesh stretching, skewness, aspect ratio and curvature in 2D) on the discretization error. For a 1D model function, mesh adaption based upon function equidistribution is explored with different relocation techniques. For 1D and 2D Burgers’ equation, mesh adaption is performed with different adaption monitors based upon truncation error and solution features. The results for two different Reynolds number are presented and discussed.