A mixed-order compact generalized finite difference method for stable seismic wavefield simulation

Abstract: In this paper, we simulated two-dimension numerical on the strong ground motion in Lanzhou basin through the hybrid scheme based on the pseudospectral method (PSM) and finite difference method (FDM). We base on a focal of 20 km deep and a profile of 5 layers is used as model to analyze the site response and the peak displacement of strong ground motion. The results show that the hybrid PSM/FDM method for seismic wavefield simulation combines with advantages of PSM and FDM and makes up for the disadvantage of them, so this method can process well the calculation of the discontinuous medium surface, then the calculation accuracy is similar to PSM. Through the wavefield simulation it is known that the peak ground displacement (PGD) of the vertical is larger and the influence of surface wave at the basin edge is more obvious than the horizontal.

Simulation of tetrahydrofuran-water hydrate formation with a combined cellular automata method and finite difference

A new mathematical treatment is proposed to the hybrid method of charge simulation and finite differences for the computation of electric fields, entirely applicable to the similar hybrid method of charge simulation and finite elements. The resulting system of linear equations is solved by making use of the fixed point theory, the QR decomposition (by using the modified Gram-Schmidt method) and the conjugate gradients squared method with a preconditioning technique. New procedures are suggested for the discretization of the boundary conditions, which lead to results with higher precision. Case studies are included.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The present thesis describes the development of a computational method for the numerical simulation of compressible flows in aerodynamics. The main goal is to study high-order discretization schemes that can potentially deliver a better computational efficiency. The spatial part of the conservation equations is discretized with the finite difference method. In particular, Summation By Parts (SBP) finite difference operators and treatment of boundary conditions via Simultaneous Approximation Terms (SAT) have been investigated. An SBP finite difference operator is essentially a (high-order) centered finite difference scheme with a specific closure at the boundary. The SAT are penalty-like terms that enforce the boundary conditions weakly and are used to augment the SBP schemes. The combination of SBP operators with SAT boundary terms constitutes schemes that, for the linearized equations governing a smooth flow, are provably stable (thus convergent) schemes. The discretization in time is carried out with high-order schemes as well. Both implicit and explicit time integration schemes have been investigated. The system of non-linear equations that arises when using the implicit time integration schemes is solved via (a damped) Newton's method. On the target grid, the initial guess required for Newton's method is obtained by performing grid sequencing combined a series of explicit relaxation iterations on each grid level. The flow equations and the equation of the turbulence model (if present) are solved in a fully-coupled manner. Numerical simulations for inviscid flow, laminar flow, as well as simulations based on DNS, LES and RANS have been performed. Results show that the described computational method is particularly well-suited for smooth external flows about relatively simple geometries, solved with the DNS and LES approach. For these types of flows the gain in efficiency obtained by using high-order schemes is clear (more than a factor of 5). For RANS, however, the comparison with a state-of-the-art second order finite volume code suggests that solving the turbulence and the flow equations fully coupled with implicit schemes may not be the most efficient iteration strategy. On the other hand, the numerical simulations of non-smooth/internal flows highlighted the need of further development of the SBP-SAT framework.

Summary We propose to combine Liao’s transmission formula(Liao et al. 1987; Liu and Sen 2010)and the staggered time integration method for the second order wave equation ( Lee, 2018 ) to absorb edge reflections in the homogeneous medium model and the inhomogenous medium model (BP gas model). Lee (2018) converted the high-order derivative of time to the calculation of the spatial derivative using the Fourier pseudospectral method,so that the numerical solution of the equation has almost no time dispersion error. And the simulation example shows that the weighted absorption boundary can absord almost all boundary reflections with a small absorption thickness. Therefore, combined with our weighted mixed absorbing edge boundary, the staggered time integration forward method may have wide applications with seismic wave field simulation, such as migration and full-waveform inversion.

A numerical comparison of finite difference (FD) and finite element (FE) methods for a stochastic ordinary differential equation is made. The stochastic ordinary differential equation is turned into a set of ordinary differential equations by applying polynomial chaos, and the FD and FE methods are then implemented. The resulting numerical solutions are all non-negative. When orthogonal polynomials are used for either continuous or discrete processes, numerical experiments also show that the FE method is more accurate and efficient than the FD method.

A high-order accurate flow solution method for quadrilateral elements is developed and tested. The method employs the flux reconstruction (FR) approach proposed by Huynh. Ideal orders of accuracy are achieved for several problems on structured-based grids while some deterioration is observed on purely unstructured grids. Cares for curved wall boundary are also shown. The localized artificial diffusivity (LAD) method is incorporated to capture discontinuities that is crucial for compressible flow simulations. The FR+LAD works very well for 1D test cases but multi-dimensional shock problems requires further modifications to confine the artificial diffusivity to the shock location. Hybrid scheme of the FR and finite difference method taking advantage of 1D-feature alleviates the instability. I. Introduction Expanding demands for the computational fluid dynamics (CFD) requires more reliable numerical schemes. Specifically, turbulent flow simulations, acoustic simulation, combustion simulations and their combined problems. The key words in such simulations may be high accuracy and high resolution. In structured grid CFD, a compact finite difference scheme satisfies both requirements at higher level than existing other methods. If we put more emphasis on easier handling of complex geometries and flow-data dependent grid generation (grid adaptation), then unstructured grid method should also be explored and it should depart from 2nd-order scheme with limiter. Recent efforts in this area are development of the discontinuous Galerkin (DG) method, 1,2 the spectral finite volume (SV) method 3 and the staggered-grid multi-domain method spectral finite difference (SG) method 4,5 and its evolutional method called a spectral difference method. 6 The common and maybe the only way for those unstructured methods is to add degrees of freedom in each cell to achieve high order since the wide stencil of the original grid is not possible due to unstructured nature. The ENO-type method that uses original degrees of freedom in the grids and extends adaptive stencils to the flow solutions is also explored but the difficulty of extending stencils due to undirectional nature of unstructured grids and the unsuitability for parallel computing prohibits wide applications. A flux reconstruction (FR) scheme 7,8 studied in the paper is the new one among the former group. It uses differential form of conservation laws like the SD method but it is more general and includes equivalent counter part of the schemes in the group. In this paper, the performance of the FR is tested by a newly developed Navier-Stokes solver. Realistic flow applications of the FR in the Navier-Stokes equations are seldom seen since it is relatively new scheme. The Localized artificial diffusivity (LAD) 9–13 is incorporated for the shock capturing in the FR. The last item is motivated by the challenge and success of combining the SD and LAD in the recent study. 11,12 Wang 14 developed the lifting collocation penalty (LCP) method to extend the FR to triangular or tetrahedral grids and applied the LCP-FR combined method for hybrid unstructured grids. In the present study, only quadrilateral cells are treated to directly apply the LAD method that was developed on structured grids. The shock capturing by the FR+LAD high order scheme is tried.

High-precision finite difference (FD) wavefield simulation is one of the key steps for the successful implementation of full-waveform inversion and reverse time migration. Most explicit FD schemes for solving seismic wave equations are not compact, which leads to difficulty and low efficiency in boundary condition treatment. Firstly, we review a family of tridiagonal compact FD (CFD) schemes of various orders and derive the corresponding optimization schemes by minimizing the error between the true and numerical wavenumber. Then, the optimized CFD (OCFD) schemes and a second-order central FD scheme are used to approximate the spatial and temporal derivatives of the 2D acoustic wave equation, respectively. The accuracy curves display that the CFD schemes are superior to the central FD schemes of the same order, and the OCFD schemes outperform the CFD schemes in certain wavenumber ranges. The dispersion analysis and a homogeneous model test indicate that increasing the upper limit of the integral function helps to reduce the spatial error but is not conducive to ensuring temporal accuracy. Furthermore, we examine the accuracy of the OCFD schemes in the wavefield modeling of complex structures using a Marmousi model. The results demonstrate that the OCFD4 schemes are capable of providing a more accurate wavefield than the CFD4 scheme when the upper limit of the integral function is 0.5π and 0.75π.

In this work, we develop a novel streamline (SL) simulation method that integrates seamlessly within the embedded discrete fracture model (EDFM). The novel SL-based method is developed based on a hybrid of two-point flux approximation (TPFA) and mimetic finite difference (MFD) methods, which is applicable to a two-phase anisotropic flow in fractured reservoirs. We refer to this novel approach as EDFM-TPFA-MFD-SL. The approach is operated in an IMplicit Pressure Explicit Saturation (IMPES) manner. First, this work establishes a novel EDFM utilizing a hybrid TPFA-MFD scheme to solve the pressure equation for phase flux approximation. Subsequently, we introduce a practical streamline tracing workflow designed to derive the distribution of streamlines within the reservoir domain and the time-of-flight distribution along each individual streamline. This feature allows for the parallel computation of water saturation along the streamlines. Two numerical examples are presented to validate the superiority of the proposed EDFM-TPFA-MFD-SL method compared to the existing streamline-based EDFM on cases with full-tensor permeability. The results show that the proposed method could significantly mitigate the numerical dissipation and reduce the computation costs. Another numerical example demonstrates the effectiveness of the proposed method in dealing with complex fracture networks and providing rapid flow diagnostics, indicating its significant potential for real-world field applications.

Recently, the finite difference method, a tool for wave acoustic simulations, can be applied to practical analysis. However, the numerical dispersion which makes propagation speed in simulations change according to the wave length, is a severe problem to maintain high accuracy in the analysis. The numerical error accompanied by the time integration hinders to carry out an accurate simulation. An examination of a compact finite difference scheme which can approximate differentiation precisely on a comparatively large grid spacing was made. It was shown that the adjusting the coefficients in the compact finite difference scheme can simulate a wave propagation accurately with a small amount of computation. The improvement method of the time integration was also discussed. The possibility of improvement of accuracy through a multi‐step integration is shown.

In this paper we present a 3D nonlinear dynamic model which describes the transient mechanical analysis of an ohmic contact RF MEMS switch, using finite element analysis in combination with the finite difference method. The model includes real switch geometry, electrostatic actuation, the two-dimensional non-uniform squeeze-film damping effect, the adherence force, and a nonlinear spring to model the interaction between the contact tip and the drain. The ambient gas in the package is assumed to act as an ideal and isothermal fluid which is modeled using the Reynolds squeeze-film equation which includes compressibility and slip flow. A nonlinear contact model has been used for modeling contact between the microswitch tip and the drain electrode during loading. The Johnson–Kendall–Roberts (JKR) contact model is utilized to calculate the adherence force during unloading. The developed model has been used to simulate the overall dynamic behavior of the MEMS switches including the switching speed, impact force and contact bounce as influenced by actuation voltage, damping, materials properties and geometry. Meanwhile, based on a simple undamped spring–mass system, a dual voltage-pulse actuation scheme, consisting of actuation voltage (Va), actuation time (ta), holding voltage (Vh) and turn-on time (ton), has been developed to improve the dynamic response of the microswitch. It is shown that the bouncing of the switch after initial contact can be eliminated and the impact force during contact can be minimized while maintaining a fast close time by using this open-loop control approach. It is also found that the dynamics of the switch are sensitive to the variations of the shape of the dual pulse scheme. This result suggests that this method may not be as effective as expected if the switch parameters such as threshold voltage, fundamental frequencies, etc. deviate too much from the design parameters. However, it is shown that the dynamic performance may be improved by increasing the damping force. The simulation results obtained from this dynamic model are confirmed by experimental measurement of the RF MEMS switches which were developed at the Northeastern University. It is anticipated that the simulation method can serve as a design tool for dynamic optimization of the microswitch. In addition, the approach of tailoring actuation voltage and the utilization of squeeze-film damping may provide further improvements in the operation of RF MEMS switches.

A coupled finite difference and material point method (CFDMP) was presented to simulate the dynamic of the gas–solid flow of thermobaric explosions. The coupled method takes advantages of both the two methods where the finite difference method captures the wave structures of gas phase and the material point method tracks the movement of solid phase. Particle collision is easily realized in MPM framework. By using a uniform grid system, the grid nodes have both the two phases’ information and their interactions are directly calculated. The simulation of thermobaric explosions in different containers is carried out and the results validate the accuracy of CFDMP program. The particle collision reduce the compression degree of the solid phase in the initial expansion stage but slightly influence the pressure history at the gauge. The process of particle movement in the container is demonstrate and the wall-reflected wave enhances the burning rate of particles.

A quantative comparison between the boundary integral equation (BIE) method and the finite difference (FD) method is presented in which each technique is applied to an elliptic boundary‐value problem (BVP) containing a boundary singularity. Two types of singularity have previously been analysed theoretically, namely those due to a discontinuous boundary potential, which we shall refer to as S1, and those due to a sudden change from the specification of boundary potential flux, an S2 singularity. In this paper the analysis is presented for a third type of boundary singularity, namely an S3 singularity: that arising from a discontinuous boundary flux. Such a condition is frequently encountered in the field of heat transfer where, for example, a system or pipe has a change of lagging material.In general, it is found that the BIE method is superior, with regards to the computational time required to achieve a certain degree of accuracy, over standard FD methods even when there is a boundary singularity. Further, the BIE method determines the solution near the singularity much more accurately than the FD method. The FD method does, however, have advantages for a very restrictive class of problems; for example, when the boundary conditions are of the Dirichlet type and the boundary geometry is rectangular. In this case an optimum relaxation parameter can easily be obtained. A soon as Neumann conditions are prescribed, the BIE is far more efficient than the FD, whatever the boundary geometry.It is concluded that, for fast, accurate solutions of general Laplacian boundary‐value problems, the BIE is appreciably superior to the FD and this is even more pronounced when there is a boundary singularity.

A finite element method (FEM) for solution of the conservative full potential flow equation is presented as a means for improving the computation of transonic flows about complex geometries. The method does not require an orthogonal mesh, thus removing a major constraint in grid generation. A standard Galerkin finite element formulation is used in conjunction with the artificial compressibility method which stabilizes the algorithm in transonic flow and allows the capture of embedded shock waves. Solution comparisons, including one test case which stresses the'capabilities of state-of-the-art finite difference methods, verify the FEM accuracy and geometric capability. I. Introduction W HILE the computation of inviscid two- and threedimensional flowfields by finite difference methods has now become commonplace in industry, one of the most important technological deficiencies is in the area of geometric modeling, particularly for transonic flows. The finite element method (FEM) is presented as a way of alleviating some problems of geometric modeling which limit the effectiveness of current finite difference methods. Certain specific advantages of the FEM, if fully exploited, may improve greatly the computational state-of-the-art. Since the governing equations of inviscid transonic potential flow are nonlinear, the method of superposition (i.e., panel methods) is insufficient to yield a proper solution, even though solutions have been obtained by a method combining surface singularities and field differences.1 Also, because of the nonlinearity, similarity laws do not exist which allow subsonic flows to be related to transonic flows. Field methods must be used, typically finite difference or finite element methods, requiring a grid of points to be defined, both on the body surface arid in the flowfield, on which solutions to the governing equations are approximated. Finite difference method (FDM) computer codes generally

A new technique for thermal field simulation of castings using a non-uniform elements method based on Finite Difference Method (FDM) is proposed. It is an improvement on the uniform elements method, which is now used in our modelling software package. In this new method, the elements in the casting/mould system are classified into two types—coarse and fine. Assumptions and principles used to distinguish coarse and fine meshes, together with the flow chart of the algorithm and a 2-D example, are given. 3-D FDM heat conductive equations of two types of mesh are deduced. The technique has been applied in practical simulation and an example is presented. Calculated results are in good accordance with practical production and those from the uniform elements method, but computing time is greatly shortened. So this non-uniform elements method has good prospects for future application and can be of benefit to the Computer Aided Engineering (CAE) technology used in the foundry industry.