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An exploration of the IGA method for efficient reservoir simulation
Novel numerical methods present exciting opportunities to improve the efficiency of reservoir simulators. Because potentially significant gains to computational speed and
accuracy may be obtained, it is worthwhile explore alternative computational algorithms
for both general and case-by-case application to the discretization of the equations of porous media flow, fluid-structure interaction, and/or production. In the present
work, the fairly new concept of isogeometric analysis (IGA) is evaluated for its suitability
to reservoir simulation via direct comparison with the industry standard finite difference (FD) method and 1st order standard finite element method (SFEM). To this end, two main studies are carried out to observe IGAâs performance with regards to geometrical modeling and ability to capture steep saturation fronts. The first study explores IGAâs ability to model complex reservoir geometries, observing L2 error convergence rates under a variety of refinement schemes. The numerical experimental setup includes an 'S' shaped line sink of varying curvature from which water is produced in a 2D homogenous domain. The accompanying study simplifies the domain to 1D, but adds in multiphase physics that traditionally introduce difficulties associated with modeling of a moving saturation front. Results overall demonstrate promise for the IGA method to be a particularly effective tool in handling geometrically difficult features while also managing typically challenging numerical phenomena.Petroleum and Geosystems Engineerin
Phase-field boundary conditions for the voxel finite cell method: surface-free stress analysis of CT-based bone structures
The voxel finite cell method employs unfitted finite element meshes and voxel quadrature rules to seamlessly
transfer CT data into patient-specific bone discretizations. The method, however, still requires the explicit
parametrization of boundary surfaces to impose traction and displacement boundary conditions, which
constitutes a potential roadblock to automation. We explore a phase-field based formulation for imposing
traction and displacement constraints in a diffuse sense. Its essential component is a diffuse geometry model
generated from metastable phase-field solutions of the Allen-Cahn problem that assumes the imaging data as
initial condition. Phase-field approximations of the boundary and its gradient are then employed to transfer
all boundary terms in the variational formulation into volumetric terms. We show that in the context of the
voxel finite cell method, diffuse boundary conditions achieve the same accuracy as boundary conditions
defined over explicit sharp surfaces, if the inherent length scales, i.e., the interface width of the phase-field,
the voxel spacing and the mesh size, are properly related. We demonstrate the flexibility of the new method
by analyzing stresses in a human femur and a vertebral body
Stochastic fiber dynamics in a spatially semi-discrete setting
We investigate a spatially discrete surrogate model for the dynamics of a
slender, elastic, inextensible fiber in turbulent flows. Deduced from a
continuous space-time beam model for which no solution theory is available, it
consists of a high-dimensional second order stochastic differential equation in
time with a nonlinear algebraic constraint and an associated Lagrange
multiplier term. We establish a suitable framework for the rigorous formulation
and analysis of the semi-discrete model and prove existence and uniqueness of a
global strong solution. The proof is based on an explicit representation of the
Lagrange multiplier and on the observation that the obtained explicit drift
term in the equation satisfies a one-sided linear growth condition on the
constraint manifold. The theoretical analysis is complemented by numerical
studies concerning the time discretization of our model. The performance of
implicit Euler-type methods can be improved when using the explicit
representation of the Lagrange multiplier to compute refined initial estimates
for the Newton method applied in each time step.Comment: 20 pages; typos removed, references adde
Colliding Interfaces in Old and New Diffuse-interface Approximations of Willmore-flow
This paper is concerned with diffuse-interface approximations of the Willmore
flow. We first present numerical results of standard diffuse-interface models
for colliding one dimensional interfaces. In such a scenario evolutions towards
interfaces with corners can occur that do not necessarily describe the adequate
sharp-interface dynamics.
We therefore propose and investigate alternative diffuse-interface
approximations that lead to a different and more regular behavior if interfaces
collide. These dynamics are derived from approximate energies that converge to
the -lower-semicontinuous envelope of the Willmore energy, which is in
general not true for the more standard Willmore approximation
Feature based volumes for implicit intersections.
The automatic generation of volumes bounding the intersection of two implicit surfaces (isosurfaces of real functions of 3D point coordinates) or feature based volumes (FBV) is presented. Such FBVs are defined by constructive operations, function normalization and offsetting. By applying various offset operations to the intersection of two surfaces, we can obtain variations in the shape of an FBV. The resulting volume can be used as a boundary for blending operations applied to two corresponding volumes, and also for visualization of feature curves and modeling of surface based structures including microstructures
Integration over curves and surfaces defined by the closest point mapping
We propose a new formulation for integrating over smooth curves and surfaces
that are described by their closest point mappings. Our method is designed for
curves and surfaces that are not defined by any explicit parameterization and
is intended to be used in combination with level set techniques. However,
contrary to the common practice with level set methods, the volume integrals
derived from our formulation coincide exactly with the surface or line
integrals that one wish to compute. We study various aspects of this
formulation and provide a geometric interpretation of this formulation in terms
of the singular values of the Jacobian matrix of the closest point mapping.
Additionally, we extend the formulation - initially derived to integrate over
manifolds of codimension one - to include integration along curves in three
dimensions. Some numerical examples using very simple discretizations are
presented to demonstrate the efficacy of the formulation.Comment: Revised the pape
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