Implementation of SIBCs for Volume Discretization Methods

2009 ◽  
pp. 263-297
Keyword(s):  
Discretization Methods

A New Global Spatial Discretization Method for Calculating Dynamic Responses of Two-Dimensional Continuous Systems With Application to a Rectangular Kirchhoff Plate

2017 ◽  
Vol 140 (1) ◽  
Author(s):  
K. Wu ◽  
W. D. Zhu
Keyword(s):  
Boundary Conditions ◽  
Continuous System ◽  
Dynamic Responses ◽  
Kirchhoff Plate ◽  
Continuous Systems ◽  
Discretization Method ◽  
Two Dimensional ◽  
Spatial Discretization ◽  
Arbitrary Boundary ◽  
Discretization Methods

A new global spatial discretization method (NGSDM) is developed to accurately calculate natural frequencies and dynamic responses of two-dimensional (2D) continuous systems such as membranes and Kirchhoff plates. The transverse displacement of a 2D continuous system is separated into a 2D internal term and a 2D boundary-induced term; the latter is interpolated from one-dimensional (1D) boundary functions that are further divided into 1D internal terms and 1D boundary-induced terms. The 2D and 1D internal terms are chosen to satisfy prescribed boundary conditions, and the 2D and 1D boundary-induced terms use additional degrees-of-freedom (DOFs) at boundaries to ensure satisfaction of all the boundary conditions. A general formulation of the method that can achieve uniform convergence is established for a 2D continuous system with an arbitrary domain shape and arbitrary boundary conditions, and it is elaborated in detail for a general rectangular Kirchhoff plate. An example of a rectangular Kirchhoff plate that has three simply supported boundaries and one free boundary with an attached Euler–Bernoulli beam is investigated using the developed method and results are compared with those from other global and local spatial discretization methods. Advantages of the new method over local spatial discretization methods are much fewer DOFs and much less computational effort, and those over the assumed modes method (AMM) are better numerical property, a faster calculation speed, and much higher accuracy in calculation of bending moments and transverse shearing forces that are related to high-order spatial derivatives of the displacement of the plate with an edge beam.

Accurate Multi-Phase Flow Simulation in Faulted Reservoirs Using Mimetic Finite Difference Methods on Polyhedral Cells

2021 ◽  
Author(s):  
Rencheng Dong ◽  
Faruk O. Alpak ◽  
Mary F. Wheeler
Keyword(s):  
Hybrid Method ◽  
Corner Point ◽  
Flow Simulation ◽  
Phase Flow ◽  
Reference Solution ◽  
Pressure Equation ◽  
Discretization Methods ◽  
Multi Phase Flow ◽  
Polyhedral Cells ◽  
Multi Phase

Abstract Faulted reservoirs are commonly modeled by corner-point grids. Since the two-point flux approximation (TPFA) method is not consistent on non-orthogonal grids, multi-phase flow simulation using TPFA on corner-point grids may have significant discretization errors if grids are not K-orthogonal. To improve the simulation accuracy, we developed a novel method where the faults are modeled by polyhedral cells, and mimetic finite difference (MFD) methods are used to solve flow equations. We use a cut-cell approach to build the mesh for faulted reservoirs. A regular orthogonal grid is first constructed,and then fault planes are added by dividing cells at fault planes. Most cells remain orthogonal while irregular non-orthogonal polyhedral cells can be formed with multiple cell divisions. We investigated three spatial discretization methods for solving the pressure equation on general polyhedral grids, including the TPFA, MFD and TPFA-MFD hybrid methods. In the TPFA-MFD hybrid method, the MFD method is only applied to part of the domain while the TPFA method is applied to rest of the domain. We compared flux accuracy between TPFA and MFD methods by solving a single-phase flow problem. The reference solution is obtained on a rectangular grid while the same problem is solved by TPFA and MFD methods on a grid with distorted cells near a fault. Fluxes computed using TPFA exhibit larger errors in the vicinity of the fault while fluxes computed using MFD are still as accurate as the reference solution. We also compared saturation accuracy of two-phase (oil and water) flow in faulted reservoirs when the pressure equation is solved by different discretization methods. Compared with the reference saturation solution, saturation exhibits non-physical errors near the fault when pressure equation is solved by the TPFA method. Since the MFD method yields accurate fluxes over general polyhedral grids, the resulting saturation solutions match the reference saturation solutions with an enhanced accuracy when the pressure equation is solved by the MFD method. Based on the results of our simulation studies, the accuracy of the TPFA-MFD hybrid method is very close to the accuracy of the MFD method while the TPFA-MFD hybrid method is computationally cheaper than the MFD method.

scholarly journals Dynamic and Sensitivity Analysis General Non-Conservative Asymmetric Mechanical Systems

2018 ◽  
Vol 68 (2) ◽  
pp. 105-124 ◽  
Author(s):  
Milan Žmindák
Keyword(s):  
Sensitivity Analysis ◽  
Modal Analysis ◽  
Dynamic Systems ◽  
Mechanical Systems ◽  
Conservative System ◽  
Eigenvalues And Eigenvectors ◽  
Linear Dynamic Systems ◽  
Discretization Methods ◽  
Linear Dynamic ◽  
Derivatives Of

AbstractIn this paper the concept of generalized form of proportional damping is proposed. Classical modal analysis of non-conservative continua is extended to multi DOF linear dynamic systems with asymmetric matrices. Mode orthogonality relationships have been generalized to non-conservative systems. Several discretization methods of continua are presented. Finally, an expression for derivatives of eigenvalues and eigenvectors of non-conservative system is presented. Examples are provided to illustrate the proposed methods.

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