Nonconforming Approach Research Articles

Simulation of coupled multiphase flow and geomechanics in porous media with embedded discrete fractures

AbstractIn fractured natural formations, the equations governing fluid flow and geomechanics are strongly coupled. Hydrodynamical properties depend on the mechanical configuration, and they are therefore difficult to accurately resolve using uncoupled methods. In recent years, significant research has focused on discretization strategies for these coupled systems, particularly in the presence of complicated fracture network geometries. In this work, we explore a finite‐volume discretization for the multiphase flow equations coupled with a finite‐element scheme for the mechanical equations. Fractures are treated as lower dimensional surfaces embedded in a background grid. Interactions are captured using the embedded discrete fracture model (EDFM) and the embedded finite element method (EFEM) for the flow and the mechanics, respectively. This nonconforming approach significantly alleviates meshing challenges. EDFM considers fractures as lower dimension finite volumes that exchange fluxes with the rock matrix cells. The EFEM method provides, instead, a local enrichment of the finite‐element space inside each matrix cell cut by a fracture element. Both the use of piecewise constant and piecewise linear enrichments are investigated. They are also compared to an extended finite element approach. One key advantage of EFEM is the element‐based nature of the enrichment, which reduces the geometric complexity of the implementation and leads to linear systems with advantageous properties. Synthetic numerical tests are presented to study the convergence and accuracy of the proposed method. It is also applied to a realistic scenario, involving a heterogeneous reservoir with a complex fracture distribution, to demonstrate its relevance for field applications.

Conforming and nonconforming laminated finite element Kirchhoff nanoplates in bending using strain gradient theory

This paper presents a comprehensive numerical finite element implementation of the nonlocal strain gradient theory applied to thin laminated composite nanoplates using Kirchhoff theory (known as Classical Laminated Plate Theory or CLPT). Hermite interpolation functions are used to approximate membrane and bending degrees of freedom according to the conforming and nonconforming approaches. To the best of the authors’ knowledge, there is no finite element formulation in the literature able to deal with laminated Kirchhoff plates including the strain gradient theory, which allows to consider general stacking sequences and boundary conditions. A simple and effective matrix notation is employed to facilitate the computer implementation. Benchmarks reported prove the accuracy of the implementation. Novel applications are provided for further developments in the subject.

A Unified Framework for the Solution of Hyperbolic PDE Systems Using High Order Direct Arbitrary-Lagrangian–Eulerian Schemes on Moving Unstructured Meshes with Topology Change

In this work, we review the family of direct Arbitrary-Lagrangian–Eulerian (ALE) finite vlume (FV) and discontinuous Galerkin (DG) schemes on moving meshes that at each time step are rearranged by explicitly allowing topology changes, in order to guarantee a robust mesh evolution even for high shear flow and very long evolution times. Two different techniques are presented: a local nonconforming approach for dealing with sliding lines, and a global regeneration of Voronoi tessellations for treating general unpredicted movements. Corresponding elements at consecutive times are connected in space-time to construct closed space-time control volumes, whose bottom and top faces may be polygons with a different number of nodes, with different neighbors, and even degenerate space-time sliver elements. Our final ALE FV-DG scheme is obtained by integrating, over these arbitrary shaped space-time control volumes, the space-time conservation formulation of the governing hyperbolic PDE system: so, we directly evolve the solution in time avoiding any remapping stage, being conservative and satisfying the GCL by construction. Arbitrary high order of accuracy in space and time is achieved through a fully discrete one-step predictor–corrector ADER approach, also integrated with well balancing techniques to further improve the accuracy and to maintain exactly even at discrete level many physical invariants of the studied system. A large set of different numerical tests has been carried out in order to check the accuracy and the robustness of our methods for both smooth and discontinuous problems, in particular in the case of vortical flows.

Mass-corrections for the conservative coupling of flow and transport on collocated meshes

Buoyancy-driven flow models demand a careful treatment of the mass-balance equation to avoid spurious source and sink terms in the non-linear coupling between flow and transport. In the context of finite-elements, it is therefore commonly proposed to employ sufficiently rich pressure spaces, containing piecewise constant shape functions to obtain local or even strong mass-conservation. In three-dimensional computations, this usually requires nonconforming approaches, special meshes or higher order velocities, which make these schemes prohibitively expensive for some applications and complicate the implementation into legacy code. In this paper, we therefore propose a lean and conservatively coupled scheme based on standard stabilized linear equal-order finite elements for the Stokes part and vertex-centered finite volumes for the energy equation. We show that in a weak mass-balance it is possible to recover exact conservation properties by a local flux-correction which can be computed efficiently on the control volume boundaries of the transport mesh. We discuss implementation aspects and demonstrate the effectiveness of the flux-correction by different two- and three-dimensional examples which are motivated by geophysical applications.

Non-conforming curved finite element schemes for time-dependent elastic–acoustic coupled problems

High-order numerical methods for solving time-dependent acoustic-elastic coupled problems are introduced. These methods, based on Finite Element techniques, allow for a flexible coupling between the fluid and the solid domain by using non-conforming meshes and curved elements.Since characteristic waves travel at different speeds through different media, specific levels of granularity for the mesh discretization are required on each domain, making impractical a possible conforming coupling in between. Advantageously, physical domains may be independently discretized in our framework due to the non-conforming feature. Consequently, an important increase in computational efficiency may be achieved compared to other implementations based on conforming techniques, namely by reducing the total number of degrees of freedom. Differently from other non-conforming approaches proposed so far, our technique is relatively simpler and requires only a geometrical adjustment at the coupling interface at a preprocessing stage, so that no extra computations are necessary during the time evolution of the simulation.On the other hand, as an advantage of using curvilinear elements, the geometry of the coupling interface between the two media of interest is faithfully represented up to the order of the scheme used. In other words, higher order schemes are in consonance with higher order approximations of the geometry. Concerning the time discretization, we analyze both explicit and implicit schemes. These schemes are energy conserving and, for the explicit case, the stability is guaranteed by a CFL condition.In order to illustrate the accuracy and convergence of these methods, a set of representative numerical tests are presented.

A mixed finite element discretisation of thin-plate splines

Thin-plate splines are a well established technique for the interpolation and smoothing of scattered data. However, the traditional formulation of the method leads to large, dense and often ill-conditioned matrices, which reduces its applicability in practice. We present a new mixed finite element formulation based on the ideas behind the mortar finite element methods. The resulting system of equations is sparse and positive definite, and its size depends only on the number of finite elements not the number of data points. References http://geopubs.wr.usgs.gov/open-file/of00-043/bathymetry/appendices.html . C. Bernardi, Y. Maday, and A. T. Patera. A new nonconforming approach to domain decomposition: the mortar element method. In H. Brezis and J.-L. Lions, editor, {{N}onlinear Partial Differential Equations and Their Applications}, pages 13--51. Paris, 1994. X. Cheng, W. Han, and H. Huang. Some mixed finite element methods for biharmonic equation. {Journal of Computational and Applied Mathematics}, 126(1-2):91--109, 2000. http://dx.doi.org/10.1016/S0377-0427(99)00342-8 . P. G. Ciarlet. {The Finite Element Method for Elliptic Problems}. North Holland, Amsterdam, 1978. P. G. Ciarlet and P. A. Raviart. A mixed finite element method for the biharmonic equation. In C. De Boor, editor, {Symposium on Mathematical Aspects of Finite Elements in Partial Differential Equations}, pages 125--143, New York, 1974. Academic Press. J. Duchon. Splines minimizing rotation-invariant semi-norms in {S}obolev spaces. In {Constructive Theory of Functions of Several Variables, Lecture Notes in Mathematics}, volume 571, pages 85--100. Springer-Verlag, Berlin, 1977. http://www.springerlink.com/content/g27671q701166031/ . R. S. Falk. Approximation of the biharmonic equation by a mixed finite element method. {SIAM Journal on Numerical Analysis}, 15(3):556--567, 1978. http://epubs.siam.org/sinum/resource/1/sjnaam/v15/i3 . M. F. Hutchinson. A stochastic estimator of the trace of the influence matrix for {L}aplacian smoothing splines. {Communications in Statistics --- Simulation and Computation}, 18(3):1059--1076, 1989. http://www.tandfonline.com/toc/lssp20/18/3 . C. Johnson and J. Pitkaranta. Analysis of some mixed finite element methods related to reduced integration. {Mathematics of Computation}, 38(158):375--400, 1982. http://www.ams.org/journals/mcom/1982-38-158/home.html . B. P. Lamichhane, S. Roberts, and L. Stals. A mixed finite element discretization of thin plate splines based on quasi-biorthogonal systems. To be submitted. P. Monk. A mixed finite element method for the biharmonic equation. {SIAM Journal on Numerical Analysis}, 24(4):737--749, 1987. http://epubs.siam.org/sinum/resource/1/sjnaam/v24/i4 . T. Ramsay. Spline smoothing over difficult regions. {Journal of Royal Statistical Society. Series B (Statistical Methodology)}, 64(2):307--319, 2002. http://onlinelibrary.wiley.com/doi/10.1111/rssb.2002.64.issue-2/issuetoc . S. Roberts, M. Hegland, and I. Altas. Approximation of a thin plate spline smoother using continuous piecewise polynomial functions. {SIAM Journal on Numerical Analysis}, 41(1):208--234, 2003. http://epubs.siam.org/sinum/resource/1/sjnaam/v41/i1 . L. R. Scott and S. Zhang. Finite element interpolation of nonsmooth functions satisfying boundary conditions. {Mathematics of Computation}, 54(190):483--493, 1990. http://www.ams.org/journals/mcom/1990-54-190/home.html . L. Stals and S. Roberts. Smoothing large data sets using discrete thin plate splines. {Computing and Visualization in Science}, 9(3):185--195, 2006. http://www.springerlink.com/content/1432-9360/9/3/ . G. Wahba. {Spline Models for Observational Data}, volume 59 of {Series in Applied Mathematic}. SIAM, Philadelphia, first edition, 1990.

A unifying process capability metric

A new economic approach to process capability assessment is presented, which differs from the commonly used engineering metrics. The proposed metric consists of two economic capability measures – the expected profit and the variation in profit of the process. This dual economic metric offers a number of significant advantages over other engineering or economic metrics used in process capability analysis. First, it is easy to understand and communicate. Second, it is based on a measure of total system performance. Third, it unifies the fraction nonconforming approach and the expected loss approach. Fourth, it reflects the underlying interest of management in knowing the expected financial performance of a process and its potential variation.

A unifying process capability metric

A new economic approach to process capability assessment is presented, which differs from the commonly used engineering metrics. The proposed metric consists of two economic capability measures – the expected profit and the variation in profit of the process. This dual economic metric offers a number of significant advantages over other engineering or economic metrics used in process capability analysis. First, it is easy to understand and communicate. Second, it is based on a measure of total system performance. Third, it unifies the fraction nonconforming approach and the expected loss approach. Fourth, it reflects the underlying interest of management in knowing the expected financial performance of a process and its potential variation.

An edge-bubble stabilized finite element method for fourth-order parabolic problems

We develop an edge-bubble stabilized finite element method for fourth-order parabolic problems. The method begins with a non-conforming approach, in which C 0 basis functions are used to approximate the coarse scale of the bulk field. Continuity of function derivatives is enforced at element edges with Lagrange multipliers. The fine-scale bulk field is approximated with higher order edge-bubbles that are held fixed over time slabs, providing for static condensation and an elimination of the multipliers. The resulting formulation shares several common features with recent non-conforming approaches based on Nitsche's method, albeit with the important difference that stability terms follow automatically from the approximation to the fine scale. As an application, we consider the problem of plane Poiseuille flow for a second-gradient fluid. Convergence studies provided for the case of steady flow indicate synchronous rates of convergence in L 2 and H 1 error norms. Some new time-dependent results for the second-gradient theory are also provided.

Animating developable surfaces using nonconforming elements

We present a new discretization for the physics-based animation of developable surfaces. Constrained to not deform at all in-plane but free to bend out-of-plane, these are an excellent approximation for many materials, including most cloth, paper, and stiffer materials. Unfortunately the conforming (geometrically continuous) discretizations used in graphics break down in this limit. Our nonconforming approach solves this problem, allowing us to simulate surfaces with zero in-plane deformation as a hard constraint. However, it produces discontinuous meshes, so we further couple this with a "ghost" conforming mesh for collision processing and rendering. We also propose a new second order accurate constrained mechanics time integration method that greatly reduces the numerical damping present in the usual first order methods used in graphics, for virtually no extra cost and sometimes significant speed-up.

Non-conforming element for accurate modelling of MEMS

In this work different modelling techniques are investigated to simulate the dynamic behaviour of slender structures on which electrostatic forces are acting. In particular, non-conforming elements are tested to model micro-mechanical devices (or MEMS) having a very large aspect ratio. These elements are constructed on linear shape functions enriched by internal second-order polynomials. As a consequence the element compatibility is not exactly satisfied, but such elements can efficiently model beam- or shell-like structures with a small number of degrees of freedom. The advantage of non-conforming elements compared to shell or beam elements is that they are volume elements and can therefore easily be combined with other volume finite elements. For micro-mechanical systems the structure must be coupled to the electrostatic domain with the so-called electro-mechanical elements that solve for the electrostatic potential and generate the electrostatic forces. This paper shows that constructing coupled electro-mechanical models for high aspect ratio systems is then greatly simplified when non-conforming finite elements are used. The theory is presented for small deformations and also for large displacements where geometric non-linearities must be accounted for. The elements proposed in this paper are based on non-conforming formulations published earlier. The efficiency of the non-conforming approach combined with specific electro-mechanical elements is highlighted in the analysis of two simple MEMS for which the pull-in voltage is computed.

A non-conforming generalization of Raviart–Thomas elements to the spherical harmonic form of the even-parity neutron transport equation

The Raviart–Thomas finite elements provide an appropriate spatial discretization of the mixed-dual form of the diffusion equation. This discretization can then be coupled to an efficient solution method. The high performances achieved by such an approach triggered research on its possible generalization to the transport equation using a spherical harmonic (or P N) angular approximation. In view of the difficulty of developing a straightforward generalization within the mixed-dual framework, we here consider 2D non-conforming (i.e., allowing interface discontinuities) finite elements coupled to the second-order form of the transport equation. This non-conforming approach keeps the mixed-dual property of the relaxation of the flux interface continuity constraint. We investigate different non-conforming elements and compare them to the well-known Lagrangian conforming elements.

Using hierarchic interpolation with mortar element method for electrical machines analysis

This work analyzes the use of the mortar element method and moving band technique to perform the movement in electrical machines. To improve the accuracy of the methods, high order hierarchic interpolation is used on the interface between moving and fixed parts. Results with first, second, and third order hierarchic interpolation are shown in order to compare conforming and nonconforming approaches.