scholarly journals A High-Order Numerical Manifold Method for Darcy Flow in Heterogeneous Porous Media

Processes ◽  
2018 ◽  
Vol 6 (8) ◽  
pp. 111 ◽  
Author(s):  
Lingfeng Zhou ◽  
Yuan Wang ◽  
Di Feng
Keyword(s):  
Porous Media ◽  
Heterogeneous Porous Media ◽  
High Order ◽  
Darcy Flow ◽  
Numerical Manifold Method ◽  
Material Interfaces ◽  
Darcy Velocity ◽  
Cover Systems ◽  
Numerical Manifold ◽  
Discontinuous Problems

One major challenge in modeling Darcy flow in heterogeneous porous media is simulating the material interfaces accurately. To overcome this defect, the refraction law is fully introduced into the numerical manifold method (NMM) as an a posteriori condition. To achieve a better accuracy of the Darcy velocity and continuous nodal velocity, a high-order weight function with a continuous nodal gradient is adopted. NMM is an advanced method with two independent cover systems, which can easily solve both continuous and discontinuous problems in a unified form. Moreover, a regular mathematical mesh, independent of the physical domain, is used in the NMM model. Compared to the conforming mesh of other numerical methods, it is more efficient and flexible. A number of numerical examples were simulated by the new NMM model, comparing the results with the original NMM model and the analytical solutions. Thereby, it is proven that the proposed method is accurate, efficient, and robust for modeling Darcy flow in heterogeneous porous media, while the refraction law is satisfied rigorously.

Novel Insight into High-Order Numerical Manifold Method Using Complex Fourier Element Shape Functions in Statics and Dynamics

2019 ◽  
Vol 11 (06) ◽  
pp. 1950058
Author(s):  
M. Malekzadeh ◽  
S. Hamzehei-Javaran ◽  
S. Shojaee
Keyword(s):  
Degrees Of Freedom ◽  
Numerical Solutions ◽  
High Order ◽  
Numerical Manifold Method ◽  
Shape Functions ◽  
Linear Complex ◽  
Statics And Dynamics ◽  
Vibration Problems ◽  
Numerical Manifold ◽  
Discontinuous Problems

In this paper, the high-order numerical manifold method (HONMM) with new complex Fourier shape functions is developed for the simulation of elastostatic and elastodynamic problems. NMM uses two separate covers which give it the ability to analyze continuous and discontinuous problems in a unified way. The new shape functions are derived using constant and linear complex Fourier shape functions. These shape functions are able to satisfy exponential and trigonometric function fields in addition to polynomial ones, unlike classic Lagrange shape functions. Compared to the Lagrange shape functions, the proposed shape functions show much more accurate results with fewer degrees of freedom. The superiority of the proposed method over the conventional HONMM in static analysis is demonstrated through a special beam example. As cases of dynamic analysis, four free and forced vibration problems are illustrated. The results of the HONMM with the use of constant and linear complex Fourier shape functions are compared with the classic HONMM results and available analytical and other numerical solutions. The results show that the proposed method, even with less number of elements, is more accurate than the classic HONMM.

ARBITRARY DISCONTINUITIES IN THE NUMERICAL MANIFOLD METHOD

2011 ◽  
Vol 08 (02) ◽  
pp. 315-347 ◽  
Author(s):  
XINMEI AN ◽  
GUOWEI MA ◽  
YONGCHANG CAI ◽  
HEHUA ZHU
Keyword(s):  
Solid Mechanics ◽  
Shear Bands ◽  
Partition Of Unity ◽  
Physical Problem ◽  
Numerical Manifold Method ◽  
Problem Domain ◽  
Material Interfaces ◽  
Cover System ◽  
Local Approximations ◽  
Numerical Manifold

An overview of modeling arbitrary discontinuities within the numerical manifold method (NMM) framework is presented. The NMM employs a dual cover system, namely mathematical covers (MCs) and physical covers (PCs), to describe a physical problem. MCs are constructed totally independent of geometries of the problem domain, over which a partition of unity is defined. PCs are the intersections of MCs and the problem domain, over which local approximations with unknowns to be determined are defined. With such a dual cover system, arbitrary discontinuities involving jumps, kinks, singularities, and other nonsmooth features can be modeled in a convenient manner by constructing special PCs and designing tailored local approximations. Several typical discontinuities in solid mechanics are discussed. Among them are the simulations of material boundaries, voids, brittle cracks, cohesive cracks, material interfaces, interface cracks, dislocations, shear bands, high gradient zones, etc.

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