draw 3d staggered grids online

Summary

We present a new numerical technique for rubberband moving ridge modelling in 3D heterogeneous media with surface topography, which is called the 3D grid method in this newspaper. This work is an extension of the 2d grid method that models P-SV wave propagation in 2D heterogeneous media. Similar to the finite-element method in the discretization of a numerical mesh, the proposed scheme is flexible in incorporating surface topography and curved interfaces; moreover information technology satisfies the free-surface boundary atmospheric condition of 3D topography naturally. The algorithm, developed from a parsimonious staggered-grid scheme, solves the problem using integral equilibrium effectually each node, instead of satisfying elastodynamic differential equations at each node equally in the conventional finite-difference method. The computational cost and memory requirements for the proposed scheme are approximately the same every bit those used by the same guild finite-difference method. In this paper, a mixed tetrahedral and parallelepiped filigree method is presented; and the numerical dispersion and stability criteria on the tetrahedral filigree method and parallelepiped filigree method are discussed in item. The proposed scheme is successfully tested confronting an analytical solution for the 3D Lamb problem and a solution of the boundary method for the diffraction of a hemispherical crater. Moreover, examples of surface-wave propagation in an elastic half-space with a semi-cylindrical trench on the surface and 3D plane-layered model are presented.

Introduction

Finite differences accept proved their usefulness in modelling the propagation of elastic moving ridge in heterogeneous media (Kelly 1976; Virieux 1986), but the fact that they are implemented using regular grids gives rising to difficulties in incorporating surface topography and curved interfaces. As an alternative to the finite-difference method, 1 tin can use a finite-element method (Smith 1975) or finite-volume method (Dormy & Tarantola 1995). Still, the finite-element method is computationally expensive and requires large corporeality of calculator memory, specially in the 3D case; and the implementation of the gratuitous-surface boundary conditions deserves to be farther studied for the finite-volume method even in the 2D case (Dormy & Tarantola 1995). In addition, the spectral element method (Komatitsch & Tromp 1999) is derived from the finite-element method to model 3D seismic wave propagation. Inclusion of topography at the free surface of an elastic medium leads to improved modelling of well-nigh-surface scattering effects, especially those in the loftier-frequency function of the wavefield; whereas modelling curved interfaces using a staircase approximation volition produce artificial diffractions, specially for the high-social club finite-departure schemes (Muir 1992). Some efforts take been made to incorporate 3D surface topography into finite-difference and other numerical discretization methods. Tessmer & Kosloff (1994) handle the 3D surface topography by mapping a rectangular filigree on to a curved grid based on a Chebychev spectral method; Ohminato & Chouet (1997) include 3D topography past discretizing the surface in a staircase so setting the Lamé coefficients to zero on the gratuitous surface and in the atmosphere; and Hestholm & Ruud (1998) introduce the free-surface conditions of 3D topography by using a local rotated coordinated arrangement at each bespeak on the surface. However, tough and wearisome efforts have to be paid to the implementation of the gratuitous-surface weather condition for all the methods mentioned. Incidentally, the curved-filigree approach (Tessmer & Kosloff 1994) is unstable for large surface curvature (Ohminato & Chouet 1997).

A numerical technique for modelling P-SV-wave propagation in 2D heterogeneous media, derived by incorporating the basic ideas of the finite-chemical element (e.chiliad. Smith 1975), finite-book (e.one thousand. Hirsch 1988) and staggered grid finite-departure (Virieux 1986) methods, has been put forward by Zhang & Liu (1999). There it was called the grid method to distinguish it from the finite-element or finite-volume methods. The method, with approximately the aforementioned memory requirements and computational price as the second-guild staggered grid scheme (Virieux 1986), is flexible in incorporating arbitrary surface topography, inner openings and irregular interfaces. Moreover, unlike the irregular grid finite-difference scheme (Zhang 1997), the free-surface weather of circuitous geometrical boundaries are satisfied naturally for this grid method. In this paper, nosotros extend the grid scheme (Zhang & Liu 1999) from 2D to 3D heterogeneous structures. The resulting 3D grid method can accurately model the 3D topography and curved interfaces by using mixed tetrahedral and parallelepiped grids. Furthermore, no extra endeavor is needed for the 3D grid method satisfying the free-surface boundary conditions of 3D topography. Thus the inclusions of surface topography and curved interfaces in 3D elastic moving ridge modelling become an easily task for the proposed numerical technique. Different the 2nd grid method, the proposed 3D grid scheme adapts the parsimonious staggered-grid scheme of Luo & Schuster (1990). Simply three displacements and three velocities need to exist stored rather than 3 velocities and six stresses as in the conventional staggered-grid scheme (e.chiliad. Virieux 1986) in each time stride. The mixed tetrahedral and parallelepiped grids can reduce the number of the grid cells (similar to the elements in the finite-element method) in the numerical mesh in contrast with pure tetrahedral grids, so that the computational cost is reduced considerably by combining two kinds of grids. Hence a mixed tetrahedral and parallelepiped grid method is presented and discussed in detail here. Furthermore, the numerical dispersion analyses on the tetrahedral grid method and parallelepiped filigree method are performed, respectively, and the stability criterion of the mixed tetrahedral and parallelepiped grid method is presented. The proposed scheme is tested against an analytical solution for the 3D Lamb problem to demonstrate the accuracy of this modelling algorithm. Moreover, the boundary method is used to benchmark the algorithm for a model with very steep surface topography. We also present examples of a 3D aeroplane-layered model and surface-moving ridge propagation in an rubberband one-half-space with a semi-cylindrical trench on the surface.

Bones Theory

Parsimonious staggered-grid formulation

The algorithm is developed through adapting the parsimonious staggered-filigree scheme. Unlike the parsimonious staggered-filigree scheme of Luo & Schuster (1990), which employs iii displacements of two time steps, nosotros apply the iii displacements and three velocities in a staggered time level. Thus, the six stresses appear only as intermediate variables that exercise not need to exist stored. The Parsimonious staggered-filigree conception that describes the rubberband wave propagation in heterogeneous media tin can be expressed, using Cartesian coordinates, as

(ane)

(2)

(3)

where i, j, thou, l= 1,…,3, we assume the summation for repeated indices; and τ _ij are the Cartesian components of the stress tensor, u _i are the components of the particle displacement, v _i are the components of the particle velocity and c _ijkl are the components of the elastic stiffness tensor. The stiffness tensor c _ijkl may contain upwards to 21 independent parameters in the 3D case, merely for an isotropic medium, two Lamé parameters λ and μ are plenty to decide the stiffness tensor:

(four)

The proposed scheme employs a staggered grid in time. The displacements are defined at integral times and the velocities are defined at one-half times. Eq. (1) is used to obtain stresses from the known displacements at integral times; eq. (two) is employed in obtaining velocities at half times from the stresses (that will not appear in the post-obit computations); and from eq. (3) we can solve displacements at the side by side integral time. Thus, the displacement field and the velocity field are updated. Hence, for the 3D example, only 3 displacements and three velocities, rather than six stresses and three velocities as in the conventional staggered-grid scheme, demand to exist stored for the proposed parsimonious staggered-grid scheme. In the following, we volition discuss the spatial discretizations of eqs (i) and (2).

Weak form of equilibrium equations

The fundamental to the proposed method is in transforming the differential equilibrium eqs (2) into a weak grade of equilibrium equations (integral equilibrium equations) effectually each node, expressed in algebraic form. The scheme is based on a 3D discretization mesh of tetrahedrons and parallelepipedons. A local mesh around an inner node P (which is constructed by all the tetrahedrons and parallelepipedons that have a common node P) is shown in Fig. i(a), where the tetrahedrons and parallelepipedons are called the grids, a basic cell of the proposed scheme. The three displacement components and the three velocity components are both defined at the nodes of the grids, and the half dozen stress components (as intermediate variables) are defined at the centres of the grids. In the following, node P in Fig. 1 is considered in detail. For each tetrahedral grid, we link the centre of the tetrahedron and the centres of the facets and the midpoints of the edges to form a spatial folded surface, represented by the fine line in Fig. 1(b), which cuts a quarter of the volume of the tetrahedron; and for each parallelepiped grid, nosotros link the heart of the parallelepipedon and the centres of the facets and the midpoints of the edges to form a spatial folded surface, represented by the fine line in Fig. 1(c), which cuts an eighth of the volume of the parallelepipedon. Thus all spatial folded surfaces, as those fatigued by the fine lines in Figs ane(b) and (c), form a polyhedron V surrounding the node P.

Figure 1.

Local mesh for the inner node P. Part (a) is a local mesh around node P, which is constructed by all the tetrahedrons and parallelepipedons that have a common node P, (b) is a typical tetrahedral grid and (c) is a typical parallelepiped filigree. The velocity and deportation components are defined at the nodes of the grids, as nodes P, A, etc. shown in (b) and (c). The stress components (as intermediate variables) are defined at the centres of the grids, represented by circles in (b) and (c).

Figure ane.

Local mesh for the inner node P. Part (a) is a local mesh around node P, which is constructed by all the tetrahedrons and parallelepipedons that have a common node P, (b) is a typical tetrahedral grid and (c) is a typical parallelepiped grid. The velocity and displacement components are defined at the nodes of the grids, as nodes P, A, etc. shown in (b) and (c). The stress components (equally intermediate variables) are defined at the centres of the grids, represented past circles in (b) and (c).

Post-obit Zhang & Liu (1999), we integrate both sides of eq. (2) over the volume within the polyhedron V. This results in

(5)

where north _j are the direction cosines of the outward-directed normals to the facets of the polyhedron. By applying the lumped mass model to the discretization system, that is to say lumping the mass of the volume inside each grid (tetrahedron or parallelepipedon) to its nodes and setting the density ρ to be zip in the inner domain of the grid, the volume integrals on the left-hand side of eq. (5) reduce to Yard _P (∂5 _i /∂t) _P . Here (∂v _i /∂t) _P is the time derivative of the velocity v _i at node P, and , where 1000 _P ^t and M _P ^p are the sums of masses of the tetrahedral and parallelepiped grids around node P, respectively. Through introducing the supposition that the stresses are homogeneous within each filigree, we tin can rewrite the surface integrals on the right-hand side of eq. (5) equally

(6)

where yard _t denotes the number of the tetrahedral grids around node P, 1000 _p the number of the parallelepiped grids effectually node P, South _t ⁵⁰ the spatial folded surface inside the lth tetrahedral grid (equally the fine solid line and dashed line shown in Fig. 1b), and Southward _p ^l the spatial folded surface inside the lth parallelepiped filigree (as the fine solid line and dashed line shown in Fig. 1c); τ _ij ^l is the stress inside the fiftythursday tetrahedral or parallelepiped grid and northward _j ^l is the management cosine of the outward-directed normals to the surface S _t ^l or S _p ^l . Eq. (half dozen) shows that the calculations of the surface integrals on the correct-hand side of eq. (5) tin be obtained by accumulating the contributions of each filigree. Information technology is noted that the aforementioned approach has been followed by the finite-volume method (Dormy & Tarantola 1995). However, they approximated the volume integrals on the left-hand side of eq. (5) equally a product of the volume within the polyhedron 5 and the function values at node P, and and so directly evaluated the surface integrals on the right-mitt side of eq. (5) with a numerical quadrature formula, which really yielded an irregular grid finite-divergence discretized version of eq. (5).

For the lth tetrahedral filigree, such equally PABC shown in Fig. 1(b), the surface integral denotes the projection of the area vector of surface Southward _t ^l in the 10 _j direction, the value of which is related to the projected surface area of the facet ABC of the tetrahedron in the ten _j management. With (c _j ^P ) _l denoting the lth surface integral for j= 1, ii, iii, we accept

(7)

where and x _j ^C , for j= 1, 2, 3, are the 3 coordinates of nodes A, B and C. Moreover, for other three nodes A, B and C, we can achieve the coordinating surface integrals, (c _j ^A ) _l , etc., by cyclic interchange of the superscripts in the order P→A→B→C→P (see Fig. 1b); however, (c _j ^A ) _l and (c _j ^C ) _l should take the inverse values of the results of eq. (seven). For the lthursday parallelepiped grid, such as PDEFGHIJ shown in Fig. 1(c), the surface integral denotes the projection of the area vector of surface S _p ^l in the x _j direction, the value of which is related to the projected area of the aeroplane GDF of the parallelepipedon in the x _j direction. With (due east _j ^P ) ₅₀ cogent the lth surface integral for j= 1, 2, 3, we have

(eight)

where x _j ^K , x _j ^D and x _j ^F , for j= i, 2, 3, are the three coordinates of nodes G, D and F. Moreover, for nodes D, East and F we tin can achieve and (e _j ^F ) _l past replacing nodes (G, D, F) with nodes (H, Due east, P), (F, D, I) or (P, Eastward, J); and nosotros have and (see Fig. 1c).

Based on higher up discussions on the lumped mass model and the contributions of each grid to the surface integrals, nosotros tin can obtain a weak form of equilibrium equations around node P (i.e. integral equilibrium equations) every bit

(ix)

Spatial derivative formulae within the grids

Owing to the fact that the displacements are defined at the nodes of the tetrahedral and parallelepiped grids (i.e. an irregular mesh), we need to introduce irregular finite-departure operators to compute the spatial derivatives of displacement components in eq. (1). For a typical tetrahedral grid PABC (see Fig. 1b), the first-order spatial derivatives of the displacements inside the tetrahedron can be obtained from the linear tetrahedral element (Zienkiewicz & Taylor 1989) as

(10)

where c _j ^P , etc. are same every bit (c _j ^P ) _fifty , etc. expressed in eq. (vii), and u _i ^P , etc. denote the ithursday displacement component at nodes P, etc.; Five denotes the book of the tetrahedron, and we have

From eq. (x) we find the assumption that the stresses are homogeneous inside each grid to be valid for the tetrahedral grids. For the parallelepiped grids, nosotros assume the stresses at the centres of the grids to stand for the homogeneous stresses within the parallelepiped grids. Thus simply showtime-order spatial derivatives of the displacements at the centres of the grids need to exist evaluated. For a typical parallelepiped grid PDEFGHIJ (see Fig. 1c), the first-social club spatial derivatives of the displacements at its centre can be obtained using a 3D eight-node isoparametric element formula (by setting the three local coordinates to nix) equally (Zienkiewicz & Taylor 1989)

(11)

where e _j ^P , etc. are same as (e _j ^P ) _l , etc. expressed in eq. (8), u _i ^P , etc. announce the ith displacement component at nodes P, etc., and W denotes the volume of the parallelepipedon.

Numerical implementations and retentiveness requirements

Calculating ∂u _i /∂x _j for i, j= ane,…,3 in each grid using u _i by eqs (10) or (11) at fourth dimension t, then substituting them into eq. (1), nosotros obtain the stresses τ _ij inside each grid at time t. Substituting stresses into the weak form of equilibrium equations of (9), we solve time derivatives of the velocities at each node, such equally (∂v _i /∂t) _P . Then velocities of all nodes at time t + Δt/2 tin be obtained using time integration. Furthermore, the displacements of all nodes at time t + Δt can be solved using time integration past eq. (3). Thus the deportation field is updated from time level t to time level t + Δt, and the velocity field is updated from fourth dimension level t − Δt/two to time level tΔt/2.

In the applied computations, the summations of the right-hand terms of eq. (9) are completed by looping all grids instead of summing all grids around each individual node. The post-obit is a detailed description: Thus nosotros tin consummate the summations of the right-hand terms of eq. (9) past looping all grids one time. In eqs (12) and (xiii) τ _ij are the stresses inside each grid and c _j ^P etc. and e _j ^P , etc. are the geometric coefficients of each grid, which are same as (c _j ^P ) _l , etc. and (e _j ^P ) _l , etc. expressed in eqs (7) and (8).

• (i)

  computer retention units are assigned to all nodes of the discretization system;

• (ii)

  for a tetrahedral grid, such as PABC, compute (for i= 1, 2, 3)

  

  (12)

• xiii

  for a parallelepiped filigree, such as PDEFGHIJ, compute (for i= one, 2, three)

  

  (13)

• (iv)

  adding the results of eqs (12) and (13) to the memory units corresponding to nodes P, A, B and C or P, D, E, F, G, H, I and J, respectively.

It is noted from eqs (10) and (12) and eqs (11) and (13) that only the volumes and 12 geometric coefficients c _j ^P , etc. or east _j ^P , etc. are demanded in the computations for tetrahedral or parallelepiped grids, and the same coefficients c _j ^P , etc. or e _j ^P >, etc. are used in both processes of obtaining stresses from displacements (eqs 10 or 11) and obtaining velocities from stresses (eqs 12 or 13). Hence, we merely need to store a grouping of geometric coefficients for many grids that are congruent. Here all congruent grids are called 1 type of grid. If we limit the number of unlike types of grids in generating the numerical mesh, the memory requirements for the geometrical coefficients will get very low. Attributable to the facts that we do non need to store coordinates for all nodes, and that other memory requirements, such every bit 3 displacements and three velocities in one time pace of all nodes, are the same every bit in the conventional parsimonious staggered-grid scheme, we tin can thus declare that retention requirements for the proposed scheme are approximately the same as those used past the same order regular-grid finite-difference method. Moreover, simply a unmarried mesh, instead of two dual lattices every bit used in irregular grid finite-difference schemes (Dormy & Tarantola 1995; Zhang 1997), is needed for the numerical implementations of the proposed scheme. From the comparisons of eq. (10) with eq. (11) and eq. (12) with eq. (13) and begetting in the listen that the geometric coefficients of the parallelepiped grids have some kind of symmetry, we can conclude that the computational cost for the parallelepiped grids is the same every bit that for the tetrahedral grids. Since the mixed tetrahedral and parallelepiped grids can utilise fewer grid cells in the discretization of a same volume in contrast with pure tetrahedral grids, the computational cost needed past the mixed tetrahedral and parallelepiped filigree method will be much less than that needed past the tetrahedral grid method.

Effigy 2.

Local mesh for the surface node P. The ABCDEFP denotes the surface topography, represented by the black line. The figure is synthetic by all the tetrahedrons that take a common node P.

Figure two.

Local mesh for the surface node P. The ABCDEFP denotes the surface topography, represented by the black line. The figure is synthetic by all the tetrahedrons that have a mutual node P.

Boundary Conditions

Only ii types of purlieus weather condition have to be considered for modelling seismic wave propagation in complex 3D heterogeneous structures: the free-surface conditions of surface topography and inner openings, and the radiation conditions for simulating a semi-infinite medium. The implementation of the gratuitous-surface conditions, specially in the presence of 3D surface topography, is difficult for the conventional finite-difference method. Nonetheless, post-obit Zhang & Liu (1999), the proposed scheme presents a new approach for treatment the complimentary-surface conditions, which results in a natural implementation of free-surface boundary conditions for a 3D complex geometrical surface. Namely, the complimentary-surface conditions are introduced by studying the equilibrium around a local domain in the vicinity of a surface, that is to extend the weak class of equilibrium equations of eq. (9) from internal to the boundaries.

It should exist noted that the parallelepiped filigree is not suitable for the discretization of the numerical mesh in the vicinity of a surface because of the assumption that the stresses are homogeneous inside the parallelepiped grids. Therefore, the numerical mesh just below a free surface is e'er constructed by the tetrahedral grids. A typical local mesh around a surface node is shown in Fig. 2, where the surface ABCDEFP, plotted past the black line, denotes the surface topography. Integrating both sides of eq. (2) over the volume surrounded by the polyhedron (which is constructed by all tetrahedrons that have a common node P), as the polyhedron shown in Fig. 2, leads to

(14)

where T is the surface topography, S is the outward surface of the polyhedron (to the exclusion of surface topography), and northward _j are the direction cosines of the outward-directed normals to the facets of the polyhedron. Following the word on eq. (5), i.e. applying the lumped mass model to the discretization system and introducing the fact that the stresses are homogeneous inside each tetrahedral filigree, we accept

(15)

where (∂v _i /∂t) _P is the fourth dimension derivative of velocity v _i at node P, One thousand _P is a quarter of the sum of the mass of all grids around node P, m _t is the number of the tetrahedral grids around node P, and τ _ij ^l and (c _j ^P ) _l have the same pregnant as those used in eq. (9). Comparison the first terms of the right-manus side of eq. (15) with those in eq. (12), we find that these terms in eq. (15) take been obtained when looping all grids to compute the summations of the correct-hand terms of eq. (ix). Furthermore, the second terms on the right-hand side of eq. (15) are equivalent to the known full loads acting on the surface T in the x _j directions for j= 1, 2, 3 (which is the initial expression of the free-surface weather). Adding all loads acting on the surface T to the retentiveness units corresponding to node P in the process of looping the grids, we can solve the fourth dimension derivatives of velocity components at node P from eq. (xv). Hence, no extra effort is needed for the implementation of the gratis-surface weather condition of complex geometrical boundaries.

For the radiation conditions, the problems arising for the proposed scheme are the same equally for the conventional finite-difference schemes (e.m. Ohminato & Chouet 1997), and there is no special handling needed. A transmission boundary condition based on a 1D feature assay (Zhang & Liu 1997) is used hither.

Numerical Analysis

Nosotros will perform a numerical assay separately for the tetrahedral grid method and the parallelepiped grid method. For the parallelepiped grid method, we discuss a homogeneous grid example when the whole mesh is synthetic using the same parallelepipedon. Hither the parallelepipedon is formed by distorting one facet (y-z plane) of a rectangular prism to making an bending θ with respect to the horizontal facet. For the tetrahedral grid method, we talk over a special inhomogeneous filigree instance when the whole mesh is synthetic by different tetrahedrons. Here the tetrahedral grids are generated through splitting each right cube into five tetrahedrons of a regular orthogonal mesh. In the following, we volition showtime discuss the dispersion relations, stability criterion and phase-velocity dispersion for the parallelepiped grid method, then those for the tetrahedral grid method are presented. Moreover, the stability of the gratis-surface conditions is studied, and the stability benchmark for the mixed tetrahedral and parallelepiped grid method is proposed.

Dispersion relation and stability criterion for the parallelepiped grid method

The dispersion relation of the parallelepiped grid method is obtained as follows:

(16)

where Δten, Δy and Δz denote the lengths of iii sides of the parallelepiped filigree, Δt is time sampling interval, ω the frequency, θ is the angle of i facet with respect to the horizontal facet, and . Here k _x , chiliad _y and yard _z are the wavenumbers in the 10, y and z directions, respectively. Eq. (16) denotes the P-moving ridge dispersion relation when 5 is the P-wave velocity, and S-moving ridge dispersion relation when 5 is the S-moving ridge velocity. For details of the derivation of eq. (sixteen) run across Appendix A.

For stability we require sinⁱⁱ(ωΔt/2) ≤ 1.0. From eq. (16) the stability criterion is given by

(17)

where is the P-wave velocity. Let Δx≥Δz and Δy≥Δz. The stability criterion of eq. (17) can exist approximately reduced to

(xviii)

From the stability criterion of eq. (18), information technology is seen that the stability of the parallelepiped grid method is independent of Southward-wave velocity and the Poisson ratio.

Phase-velocity dispersion of the parallelepiped filigree method

Consider a plane wave with wavelength λ, for which the propagation management has management cosines T _ane, T _ii and T ₃. We have and . Let . Following Virieux (1986), define the parameters H=h/λ and . From eq. (16) the ratio of the stage velocity λω/2π to the truthful wave velocity are given by

(19)

where , and q denotes the not-dimensional P-wave stage velocity (q _p ) when η= 1 and the not-dimensional S-wave phase velocity (q _south ) when . Here υ is the Poisson ratio. The role q(H) shows, quantitatively, the phase-velocity dispersion arising from replacing the true partial differential equations with the finite-difference equations. For different H (H controls the number of nodes per wavelength), deviations of the quantities q _p and q _s from 1.0 show the degree of dispersion under this numerical mesh. Hence, the to a higher place dispersion analysis provides a rule to determine the spatial discretized size. For γ= 0.eight and θ= 60°, the P-wave phase-velocity dispersion curves, q _p (H), are plotted in Fig. three for different propagation directions of the plane wave, which are T _i= ane.0, T ₂= 0.0 and and , and and . The figure is valid for any Poisson ratio. The quantity q _p is ever lower than 1 and approaches i for modest H. For . This ways that the edges of the parallelepiped grids should be 1/20 of the wavelength in gild to model the P wave accurately. For γ= 0.viii and θ= lx°, the South-wave phase-velocity dispersion curves, q _s (H), are plotted in Fig. 4, and the direction cosines of the propagation directions of the airplane moving ridge are and and , and and . For . This gives a same dominion of thumb every bit for the P-wave modelling. Moreover, the behaviour of >q _s does not degrade as υ goes to 0.5. This suggests that the parallelepiped filigree method behaves correctly inside liquids, and at liquid-solid interfaces.

Figure 3.

Dispersion curves of the parallelepiped grid method for non-dimensional P-wave phase velocity with the parameter γ= 0.8. Results for diverse propagation directions of the aeroplane moving ridge, that are: (a) and T ₃= 0.0, (b) and , and (c) and . They are independent of the Poisson ratio.

Effigy three.

Dispersion curves of the parallelepiped filigree method for non-dimensional P-wave phase velocity with the parameter γ= 0.8. Results for diverse propagation directions of the airplane moving ridge, that are: (a) and T ₃= 0.0, (b) and , and (c) and . They are independent of the Poisson ratio.

Figure 4.

Dispersion curves of the parallelepiped filigree method for a non-dimensional Due south-wave phase velocity with γ= 0.8. Results for various propagation directions of the airplane wave: (a) and T _iii= 0.0, (b) and , and (c) and , are shown on the same graph for different Poisson ratios.

Effigy iv.

Dispersion curves of the parallelepiped grid method for a not-dimensional Southward-wave phase velocity with γ= 0.8. Results for various propagation directions of the plane wave: (a) and T ₃= 0.0, (b) and , and (c) and , are shown on the aforementioned graph for different Poisson ratios.

Dispersion relations and stability benchmark for the tetrahedral filigree method

The dispersion relations of the tetrahedral grid method for a Poisson ratio of 0.25 are obtained in Appendix B. For a special case, that is , nosotros have a simple form equally follows:

(twenty)

(21)

where h is the interval of the regular orthogonal mesh, Δt is the fourth dimension sampling interval, and and are the P- and Due south-wave velocities, respectively. Eq. (20) denotes the P-wave dispersion relation; and eq. (21) the South-wave dispersion relation. For stability we crave that . From eqs (twenty) and (21) the stability criterion can exist approximately given by

(22)

It is noted that the stability criterion of eq. (22) is merely valid for the Poisson ratio υ= 0.25. All the same, comparing eq. (22) with eq. (18) reveals that the stability criterion of the tetrahedral filigree method is more relaxed than that of the parallelepiped grid method. Thus, for the mixed tetrahedral and parallelepiped grid method presented in this newspaper, eq. (18) can serve equally a stability criterion to chose proper temporal and spatial discretized intervals.

Phase-velocity dispersion of the tetrahedral grid method

Following the give-and-take on the parallelepiped grid method, we can obtained phase-velocity dispersion curves q _p (H) and q _s (H) based on eqs (20) and (21), which are plotted in Fig. 5 for γ= 0.8, υ= 0.25 and the propagation direction of and . Information technology is found that q _p ≈ 1.0 and q _s ≈ 1.0 for H≤ 0.09 from Fig. 5. Hence, the rule of thumb, to determine the discretized size, for the parallelepiped grid method is more restrictive than that for the tetrahedral grid method. Thus, for the mixed tetrahedral and parallelepiped grid method presented in this newspaper, we can use the rule of pollex as for the parallelepiped grid method to determine the spatial discretized size.

Figure 5.

Dispersion curves of the tetrahedral grid method for non-dimensional P- and Southward-wave stage velocities with γ= 0.eight. The propagation directions of the plane moving ridge is and . The Poisson ratio is 0.25.

Figure 5.

Dispersion curves of the tetrahedral grid method for not-dimensional P- and S-wave phase velocities with γ= 0.8. The propagation directions of the aeroplane wave is and . The Poisson ratio is 0.25.

Stability of complimentary-surface conditions

It has been observed that instabilities occurring when solving hyperbolic equations are oft caused by the treatment of the free-surface boundary conditions (Gottlieb 1982; Kosloff 1990). Hence, it is interesting to analyse the stability of the free-surface conditions satisfied naturally by the 3D grid method. Nosotros study a aeroplane surface (parallel to the 10-y plane) case when the tetrahedral grids are generated through splitting each correct cube into v tetrahedrons of a regular orthogonal mesh. Following the word on the finite-divergence equations for the tetrahedral grid method, nosotros tin can achieve the finite-difference equations for the surface case (run into Appendix C). Let a airplane P wave vertically incident upon the gratis surface be . The displacement components tin can be expressed as

(23)

Calculating the displacement components at nodes from eq. (23) and then substituting them into the finite-departure equations of Appendix C, we obtain the dispersion relation for the local surface for a vertical incident P wave as follows:

(24)

From eq. (24) we can approximately solve the stability criterion for the free-surface conditions as

(25)

The stability benchmark in eq. (25) is more relaxed than that expressed in eq. (xviii) for the mixed tetrahedral and parallelepiped filigree method, so that the stability of complimentary-surface conditions tin be guaranteed for the 3D filigree method.

Numerical Simulations

Analytic comparison

The accuracy of the proposed method is tested through a comparison of numerical results with an analytical solution of the 3D Lamb trouble. The analytical solution is achieved by convolving the gratuitous-surface Green function (Pekeris 1955) with the source function. A vertical Gaussian bespeak source, with a maximum frequency of 10 Hz, is loaded at the free surface. Two numerical results, which differ in the discretization of the numerical mesh of the half-space, are illustrated in Figs 6 and 7. Fig. 6 shows a comparison of the vertical displacements at three stations, 1008, 1504 and 2000 chiliad from the source, on the free surface. The numerical model for this results is made up of tetrahedral and parallelepiped grids with the edges of the tetrahedrons and parallelepipedons of 16 yard, where the parallelepipedon is a baloney of a correct cube with its ane facet making an bending of 60° with respect to the horizontal facet and the tetrahedron is formed by splitting each of these parallelepipedons into six tetrahedrons. Fig. 7 shows a comparing of the vertical displacements for longer propagation paths at iv stations, 1200, 1800, 2400 and 3000 chiliad from the source, on the complimentary surface. The numerical model for Fig. 7 is made upwards of tetrahedral and right cube grids with a minimum border of xx grand, where the tetrahedral grids in the vicinity of the surface are generated through splitting each right cube into five tetrahedrons. This can exist considered as one kind of inhomogeneous filigree case. The computational cost of the consequence of Fig. 7 is reduced to a quarter past exploiting the symmetry of the model on xy- and yz-planes with z-axis pointing downward (which also represents another advantage of the proposed method, i.e. the symmetry condition can be implemented very easily). The receivers for the upshot of Fig. seven are positioned at a line on the surface, which starts from the source point and make an angle of 45° with respect to the x-axis. The semi-finite medium has a P-wave velocity of 3000 g s⁻¹ and an South-wave velocity of 1730 m south⁻¹ for two numerical results. The grid edges are 1/19 of P wavelength and one/11 of South wavelength for Fig. half-dozen and 1/15 of P wavelength and one/9 of S wavelength for Fig. 7. From Figs 6 and 7 we find that the surface moving ridge propagates without dispersion, and the discrepancy in amplitude between numerical and analytical Rayleigh waves is less than five per cent. The total propagation times are two.0 south with a fourth dimension pace of 3 ms for Fig. 6 and two.five southward with a time step of 4 ms for Fig. vii.

Figure six.

Comparing betwixt numerical and belittling vertical components of the deportation for the 3D Lamb problem at various stations on the gratis surface. The display is for the edge length of grids of 16 k.

Figure 6.

Comparing between numerical and analytical vertical components of the displacement for the 3D Lamb trouble at various stations on the free surface. The display is for the border length of grids of xvi thou.

Figure seven.

Comparison between numerical and analytical vertical components of the displacement at diverse stations on the free surface for longer propagation paths of the 3D Lamb trouble. The display is for an inhomogeneous grid case with a minimum border length of grids of 20 thou.

Figure 7.

Comparison between numerical and belittling vertical components of the deportation at various stations on the gratuitous surface for longer propagation paths of the 3D Lamb problem. The display is for an inhomogeneous grid example with a minimum edge length of grids of twenty m.

Layered model

A constructed seismogram is generated for a aeroplane-layered model, as illustrated in Fig. eight, to assess the proposed algorithm in modelling elastic wave propagation in heterogeneous media. A vertical Ricker wavelet signal source, with a maximum frequency of 15 Hz, is loaded at the free surface of the 3D layered model. Let the source point be , and the z-axis point vertical down. The receivers with a spacing of 14.14 m are positioned at the line on the free surface, which starts from the source betoken and make an angle of 45° against x-axis. The numerical seismogram of vertical displacements is showed in Fig. 9, where all amplitudes take their inverse values and the amplitudes of the directly and Rayleigh waves are reduced to 20 per cent in guild to show the reflected waves coming from the plane interfaces clearly. The reflected P waves coming from the first interface as a issue of the incident P wave (P-P) and the incident S wave (S-P), and reflected P moving ridge (P-P-P) coming from the second interface are seen conspicuously in Fig. 9. The practical computation is washed on a quarter of the model by exploiting the symmetry, thus the computational cost is much reduced. The numerical model is made upwardly of tetrahedral and parallelepiped grids with a minimum edge of 10 m. The time interval used is two ms.

Effigy 8.

Plane-layered model with vertically increasing velocities and density.

Figure 8.

Plane-layered model with vertically increasing velocities and density.

Effigy nine.

Numerical seismogram of vertical displacements at the free surface. The receivers are positioned at a line on the free surface, which starts from the source indicate and brand an angle of 45° against ten-axis. All amplitudes take their changed values and the amplitudes of the straight and Rayleigh waves are reduced to 20 per cent.

Figure ix.

Numerical seismogram of vertical displacements at the free surface. The receivers are positioned at a line on the free surface, which starts from the source point and make an bending of 45° against 10-centrality. All amplitudes take their inverse values and the amplitudes of the directly and Rayleigh waves are reduced to twenty per cent.

To assess the accuracy of the proposed scheme, we transform the above betoken-source response into line-source responses by carrying out an integration along the receiver coordinate (Wapenaar 1992), and and then compare the resulting line-source responses with second finite-difference modelling results. Figs 10 and 11 bear witness the comparisons of vertical displacements at stations, 40 × 14.fourteen grand and 80 × 14.fourteen 1000 from the source, on the free surface. In spite of the errors derived from the transformation of the point-source response into the line-source ane, 2 results agree well in Figs ten and 11.

Figure 10.

Comparing between the 2D finite-deviation solution with the line-source response for the vertical displacement on the free surface at a source-receiver distance of 40 × fourteen.14 m. The line-source response is obtained from the superposition of the 3D signal-source responses of Fig. 9.

Figure 10.

Comparison between the 2D finite-difference solution with the line-source response for the vertical displacement on the costless surface at a source-receiver altitude of 40 × 14.14 one thousand. The line-source response is obtained from the superposition of the 3D betoken-source responses of Fig. ix.

Figure 11.

Comparing between the 2nd finite-difference solution with the line-source response for the vertical displacement on the free surface at a source-receiver altitude of 80 × 14.xiv yard. The line-source response is obtained from the superposition of the 3D signal-source responses of Fig. ix.

Effigy 11.

Comparison between the 2D finite-deviation solution with the line-source response for the vertical deportation on the free surface at a source-receiver distance of eighty × 14.14 yard. The line-source response is obtained from the superposition of the 3D signal-source responses of Fig. 9.

Trench on the surface

To examination the quality of the proposed scheme in accounting for the surface topography, we present an example of elastic wave propagation in a semi-infinite medium with a semi-cylinder trench on the gratuitous surface. The model has a size of 1050 × 1050 × 520 k³ as shown in Fig. 12. The semi-cylindrical trench has a radius of xiv m, and it lies parallel to the x-direction. A vertical Ricker wavelet point source, with a maximum frequency of 35 Hz, is loaded at the free surface , as indicated by an arrow in Fig. 12. The normal distance between the source and the trench is 280 m. The semi-finite medium has a P-wave velocity of 2000 m s⁻¹ and an S-wave velocity of 1130 m south⁻ⁱ. Based on the symmetry of the model on the yz-airplane at 10= 0, we can reduce the computational cost past using one-half of the model in practical computations. The numerical mesh is made upwardly of tetrahedral and parallelepiped grids with a minimum edge of iii.5 m. The minimum filigree edges are one/16 the P wavelength and 1/9 the South wavelength. For details on the generation of complicated numerical mesh run across Thompson (1985). The computation, with 6 859 779 nodes, ii 614 950 tetrahedral grids and 6 300 000 parallelepiped grids, takes a CPU fourth dimension of 340 min on a Dominicus Solaris Sparc 2 when performing 450 fourth dimension steps of i ms.

Figure 12.

An elastic one-half-space with a semi-cylinder trench on the costless surface. The source, shown past an arrow, is load at the centre bespeak (O) on the surface. The normal distance betwixt the source and the trench is 280 m. The semi-cylindrical trench has a radius of xiv m. The semi-finite medium has a P-wave velocity of 2000 m s⁻¹ and an S-wave velocity of 1130 chiliad south⁻¹.

Effigy 12.

An rubberband half-space with a semi-cylinder trench on the free surface. The source, shown past an arrow, is load at the centre signal (O) on the surface. The normal altitude between the source and the trench is 280 m. The semi-cylindrical trench has a radius of 14 m. The semi-finite medium has a P-wave velocity of 2000 m s⁻¹ and an S-wave velocity of 1130 m southward⁻¹.

Snapshots of the y-direction component of the deportation on the costless surface (z= 0) and the vertical airplane (10= 0) that is orthogonal with the trench at propagation times of 0.26, 0.four and 0.45 s are shown in Fig. 13. The snapshots bear witness very clear moving ridge fronts of the Rayleigh surface wave, the direct P wave, the straight S wave, and reflected and diffracted waves acquired by the trench. The conical wave tin exist plant in the vertical plane (x= 0). Comparing the snapshots of the surface with that of the vertical plane illustrates the 'surface' characteristics of the Rayleigh moving ridge. Owing to its shorter wavelength, the diffraction of the Rayleigh wave is much stronger than that of the P wave when propagating through the trench, as tin be seen clearly in the snapshots.

Figure 13.

Snapshots of y-direction component of the deportation at propagation times of 0.26, 0.iv and 0.45 southward. The top of each figure is related to the free surface, and the bottom is related to a vertical plane (x= 0) that is orthogonal with the trench on the surface. A vertical Ricker wavelet point source is positioned at the midpoint of the intersection line of ii planes (). The trench tin can be seen in z= 0 surface at y= 280 m.

Figure 13.

Snapshots of y-direction component of the deportation at propagation times of 0.26, 0.4 and 0.45 s. The top of each figure is related to the free surface, and the bottom is related to a vertical plane (ten= 0) that is orthogonal with the trench on the surface. A vertical Ricker wavelet point source is positioned at the midpoint of the intersection line of 2 planes (). The trench tin be seen in z= 0 surface at y= 280 grand.

Hemispherical crater

An example of rubberband moving ridge propagation in a hemispherical crater in a homogeneous half-space is used to test the accurateness of the proposed algorithm in the presence of 3D surface topography. Sanchez-Sesma (1983) studied the response of a hemispherical crater in a homogeneous half-space to a vertically incident plane P wave based on an judge boundary method. He presented the amplitude of the deportation recorded at the surface of the crater for different normalized frequencies η= 2a/λ _p , where a denotes the radius of the crater and λ _p the wavelength of the incident P wave. His result is used here to test the algorithm quantitatively.

The semi-finite medium has a P-wave velocity of 3000 m s⁻ⁱ and an South-wave velocity of 1730 m southward^−one. The density is 2500 kg thou^−three. The source is a vertically incident airplane P wave, which is a Ricker wavelet in time with a peak frequency of 5 Hz. The radius of the hemispherical crater is 75 and 150 m for the normalized frequencies of 0.25 and 0.five, respectively. Transforming the seismograms for receivers at the surface into frequency domain and picking the amplitudes corresponding to v Hz, nosotros can and then compare the results with Sanchez-Sesma (1983).

The mesh is fabricated up of tetrahedral grids in the vicinity of the surface and right cube grids beneath those. Again the symmetry of the model on xy- and yz-planes with the z-axis pointing down is exploited to reduce the computational toll. Fig. fourteen shows a local mesh in the vicinity of the crater for η= 0.25. It tin can be seen that the mesh accurately models the 3D surface topography. Ordinarily the number of points per wavelength is less for college-society schemes. However, for the correct description of the geometry of the hemisphere, small mesh spacing is required even for higher-order schemes (e.yard. Komatitsch & Tromp 1999). From Fig. 14 nosotros tin over again notice the flexibility of the proposed tetrahedral and parallelepiped grid scheme, i.due east. no much extra points are needed for including surface topography and the edge of grids are more or less constant.

Figure 14.

A local mesh in the vicinity of the crater for η= 0.25. The mesh is synthetic by the tetrahedral grids.

Figure xiv.

A local mesh in the vicinity of the crater for η= 0.25. The mesh is constructed by the tetrahedral grids.

Fig. 15 shows the results of the proposed grid scheme for η= 0.25 and 0.v. The corresponding results of Sanchez-Sesma can be seen in Fig. 15 of Komatitsch & Tromp (1999). Information technology is found through comparing the two figures that the agreement is splendid. In particular, the stiff amplification close to the edges of the crater is reproduced well. Notation that the distension level of the vertical component reaches a very high value in the center for η= 0.5 (which is 2 for the airplane surface).

Figure fifteen.

Amplitudes of the two components of displacement recorded along the crater, from x/a= 0 (centre of the crater) to ten/a= 2. The vertical and radial components are displayed. The results are shown for two normalized frequency η= 0.25 (top) and η= 0.5 (bottom).

Effigy 15.

Amplitudes of the two components of displacement recorded along the crater, from x/a= 0 (centre of the crater) to x/a= ii. The vertical and radial components are displayed. The results are shown for two normalized frequency η= 0.25 (top) and η= 0.5 (bottom).

Conclusions

Nosotros accept presented a new numerical modelling algorithm for elastic wave propagation in 3D heterogeneous media. The scheme is an extension of the 2D grid method. The algorithm can accurately model the surface topography, inner openings and curved interfaces by using an unstructured mesh. The mesh used is similar to the mesh generated for the finite-chemical element method, thus many available mesh generation algorithms tin can be applied for the scheme. Moreover, no extra effort is needed for the scheme to satisfy the free-surface boundary conditions of 3D topography. The computational toll and retentivity requirements for the algorithm are approximately the same equally those used past a same gild regular-filigree finite-difference method. The properties outlined above allow the proposed scheme to easily solve the difficulties, such as the inclusion of 3D surface topography, that arise for the standard finite-difference schemes. Comparisons with an analytic solution of the 3D Lamb problem and a solution of the boundary method for the diffraction of a hemispherical crater and a 3D layered model evidence the new scheme to be highly authentic, and the numerical simulation on wave propagation in a one-half-space with a corrugated surface demonstrates its high quality. The scheme can serve every bit a powerful tool for the study of moving ridge propagation phenomena in the vicinity of non-planar surface and interfaces of complicated 3D heterogeneous structures.

Acknowledgments

We are grateful to the Pedagogy Ministry of China and the National Natural Scientific discipline Fund for Distinguished Young Scholars of China who supported this work. Nosotros thank Jeroen Tromp and an anonymous reviewer for constructive comments that improved the manuscript.

Appendix

Dispersion Relation of the Parallelepiped Grid Method

We presume a uniform infinite medium that supports a plane wave. Substituting eq. (11) into eq. (i) then into eq. (9) (together with eq. iii) provides a second-order system of difference equations in displacements only. This system tin exist written in matrix form using the second-order finite-difference operators and D _tt :

(A1)

where is the South-wave velocity and is the P-wave velocity. With p _{l,chiliad,northward} ^j denoting the field variable value at grid bespeak (fifty, 1000, n) at time level j, the second-club finite-difference operators in eq. (A1) can be expressed equally follows:

(A2)

where θ is the bending of one facet against the horizontal facet of the parallelepiped grids, Δx, Δy and Δz are the lengths of three sides of the parallelepiped grid, Δt is time sampling interval, and nosotros take

Let the frequency of the plane wave be ω and the wavenumbers in the x, y and z directions exist, respectively, one thousand _x , g _y and k _z . The displacement vector at grid point (50, m, north) at fourth dimension level j is

(A3)

where {a b c}^T is the displacement vector at (0, 0, 0) at the initial time level and . Substituting eq. (A3) into the finite-departure operators of eq. (A2) and and then substituting them into eq. (A1), we transform the coefficient matrix of eq. (A1) into a new one where the operators and D _tt are replaced by follows:

It should be noted that there should exist a non-goose egg solution for the vector {a b c}^T. Hence, the new coefficient matrix should be singular. Letting the norm of the matrix be goose egg gives the dispersion relation every bit

(A4)

Eq. (A4) denotes the P-wave dispersion relation when 5 is the P-wave velocity, and the S-wave dispersion relation when V is the S-moving ridge velocity.

Appendix

Dispersion Relations of the Tetrahedral Filigree Method

We discuss a case when the tetrahedral grids are generated through splitting each correct cube into five tetrahedrons of a regular orthogonal mesh. This inhomogeneous tetrahedral mesh makes the dispersion relations very complex. For simplicity, we assume the space medium with a Poisson ratio of 0.25, i.due east. λ=μ. Past substituting eq. (ten) into eq. (1) then into eq. (9) (together with eq. 3), the finite-departure equations for the tetrahedral grid method can be obtained as

(B1)

(B2)

(B3)

where (u _i ) _l,yard,n ^j , for i= 1,…,3, announce the displacement components at grid point (l, m, n) at time level j, Δt is time sampling interval, and we have

Here h is the interval of the regular orthogonal mesh, α and θ are the P- and S-moving ridge velocities, respectively.

Assuming a uniform infinite medium that supports a airplane wave with a frequency of ω, the displacement components at the nodes can be expressed every bit

(B4)

where m _x , chiliad _y and thousand _z are the wavenumbers in the x, y and z directions, {a b c}^T is the displacement vector at (0, 0, 0) at the initial time level. Let υ = 0.25. Substituting eq. (B4) into eqs (B1)–(B3) gives the following:

(B5)

where

It is noted that there should be a non-nil solution for the vector {a b c}^T in eq. (B5). Hence, the matrix in eq. (B5) must be singular. Substituting s ₀, etc. into the coefficient matrix of eq. (B5) and letting the norm of the matrix be nothing, we can obtain the dispersion relations for the tetrahedral grid method. These dispersion relations are very complicated. For a special case, that is , we have a simple form as follows:

(B6)

(B7)

Eq. (B7) are ii equal roots of the zero-norm of the matrix of eq. (B5), which denotes the South-wave dispersion relation; and eq. (B6) denotes the P-moving ridge dispersion relation.

Appendix

Finite-difference equations for the surface

With (u _i ) _l,m,north ^j , for i= 1,…,three, denoting the displacement components at grid indicate (l, m, northward) at time level j, the finite-difference equations for the surface can be expressed as

(C1)

(C2)

(C3)

where Δt is time sampling interval, and we have

Hither h is the interval of the regular orthogonal mesh, α and θ are the P- and Southward-wave velocities, respectively.

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