This paper provides a tutorial and survey of methods for parameterizing surfaces with a view to applications in geometric modelling and computer graphics. We gather various concepts from differential geometry which are relevant to surface mapping and use them to understand the strengths and weaknesses of the many methods for parameterizing piecewise linear surfaces and their relationship to one another.

/pdf/surface-parameterization-a-tutorial-and-survey-3u3e0o47hm.pdf

Surface Parameterization: a Tutorial and Survey

Non-Darcy behavior is important for describing fluid flow in porous media in situations where high velocity occurs. A criterion to identify the beginning of non-Darcy flow is needed. Two types of criteria, the Reynolds number and the Forchheimer number, have been used in the past for identifying the beginning of non-Darcy flow. Because each of these criteria has different versions of definitions, consistent results cannot be achieved. Based on a review of previous work, the Forchheimer number is revised and recommended here as a criterion for identifying non-Darcy flow in porous media. Physically, this revised Forchheimer number has the advantage of clear meaning and wide applicability. It equals the ratio of pressure drop caused by liquid–solid interactions to that by viscous resistance. It is directly related to the non-Darcy effect. Forchheimer numbers are experimentally determined for nitrogen flow in Dakota sandstone, Indiana limestone and Berea sandstone at flowrates varying four orders of magnitude. These results indicate that superficial velocity in the rocks increases non-linearly with the Forchheimer number. The critical Forchheimer number for non-Darcy flow is expressed in terms of the critical non-Darcy effect. Considering a 10% non-Darcy effect, the critical Forchheimer number would be 0.11.

A Criterion for Non-Darcy Flow in Porous Media

Mesh parameterization is a powerful geometry processing tool with numerous computer graphics applications, from texture mapping to animation transfer. This course outlines its mathematical foundations, describes recent methods for parameterizing meshes over various domains, discusses emerging tools like global parameterization and inter-surface mapping, and demonstrates a variety of parameterization applications.

https://hal.inria.fr/inria-00186795/document

Mesh parameterization: theory and practice

Mesh parameterization: theory and practice Video files associated with this course are available from the citation page

Numerical simulation of fluid flow and transport processes in the subsurface must account for the presence of wells. The pressure at a gridblock that contains a well is different from the average pressure in that block and different from the flowing bottom hole pressure for the well [17]. Various finite difference well models have been developed to account for the difference. This paper presents a systematical derivation of well models for other numerical methods such as standard finite element, control volume finite element, and mixed finite element methods. Numerical results for a simple well example illustrating local grid refinement effects are given to validate these well models. The well models have particular applications to groundwater hydrology and petroleum reservoirs.

Well flow models for various numerical methods

Variational method for untangling and optimization of spatial meshes

A gridgeneration method based on the minimization of a barrier functional with a feasible set consisting of quasi-isometric grids is presented. The procedure for minimizing the deviation from isometry for a grid of given connectivity and for given boundary grid points makes it possible to contract the feasible set into a small neighborhood of the optimal solution. Since the functional is defined in terms of metrics in both physical and logical coordinates, the adaptation problem can be analyzed within the framework of quasi-isometric grids. A reliable and efficient technique for constructing a feasible solution is proposed, based on the penalty formulation and continuation technique. Numerical experiments demonstrate the high quality of the generated grids.

Barrier method for quasi-isometric grid generation

Parallel versions of a method based on reducing a linear program (LP) to an unconstrained maximization of a concave differentiable piecewise quadratic function are proposed. The maximization problem is solved using the generalized Newton method. The parallel method is implemented in C using the MPI library for interprocessor data exchange. Computations were performed on the parallel cluster MVC-6000IM. Large-scale LPs with several millions of variables and several hundreds of thousands of constraints were solved. Results of uniprocessor and multiprocessor computations are presented.

Parallel implementation of Newton's method for solving large-scale linear programs

Mapping a triangulated surface to 2D space (or a tetrahedral mesh to 3D space) is an important problem in geometry processing. In computational physics, untangling plays an important role in mesh generation: it takes a mesh as an input, and moves the vertices to get rid of foldovers. In fact, mesh untangling can be considered as a special case of mapping where the geometry of the object is to be defined in the map space and the geometric domain is not explicit, supposing that each element is regular. In this paper, we propose a mapping method inspired by the untangling problem and compare its performance to the state of the art. The main advantage of our method is that the untangling aims at producing locally injective maps, which is the major challenge of mapping. In practice, our method produces locally injective maps in very difficult settings, both in 2D and 3D. We demonstrate it on a large reference database as well as on more difficult stress tests. For a better reproducibility, we publish the code in Python for a basic evaluation, and in C++ for more advanced applications.

Foldover-free maps in 50 lines of code

We describe discrete well models for 2-D non-Darcy fluid flow in anisotropic porous media Attention is mostly paid to the well models and simplified calibration procedures for the control volume mixed finite element methods, including the case of highly distorted grids

V. A. Garanzha

Papers

Variational method for untangling and optimization of spatial meshes

Barrier method for quasi-isometric grid generation

Parallel implementation of Newton's method for solving large-scale linear programs

Foldover-free maps in 50 lines of code

Validation of Non-darcy Well Models Using Direct Numerical Simulation