Citation for published version (APA): Kole, J. S. (2003). New methods for the numerical solution of Maxwell's equations s.n.

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1 University of Groningen New methods for the numerical solution of Maxwell's equations Kole, Joost Sebastiaan IMPORTANT NOTE: You are advised to consult the publisher's version publisher's PDF if you wish to cite from it. Please check the document version below. Document Version Publisher's PDF, also known as Version of record Publication date: 3 Link to publication in University of Groningen/UMCG research database Citation for published version APA: Kole, J. S. 3. New methods for the numerical solution of Maxwell's equations s.n. Copyright Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the authors and/or copyright holders, unless the work is under an open content license like Creative Commons. Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Downloaded from the University of Groningen/UMCG research database Pure: For technical reasons the number of authors shown on this cover page is limited to maximum. Download date: --9

2 Chapter 6 Elastodynamics Extension to other wavefield phenomena In the previous chapters, we have focused on solving the time-dependent Maxwell Equations. The time-dependent Maxwell Equations are a set of first-order partial differential equations, and the underlying skew-symmetry of these equations were exploited to construct unconditionally stable algorithms. However, underlying Hermitian or skew-hermitian symmetries are found in many almost all physical phenomena.therefore, the procedure to construct unconditionally algorithms to solve the Maxwell Equations, might be applicable for other cases than electromagnetism. We will extend the procedure to construct unconditionally stable algorithms to one of the most complicated linear wave equations: elastodynamics.for completeness, we briefly recall the procedure to anti-symmetrize the Maxwell Equations, and repeat this for acoustics and elastodynamics see also [87].The implementation, while conserving the skew-symmetry during spatial differentiation and time integration, will be carried out only for elastodynamics.finally, results for some typical seismic events are presented. 6. Skew-symmetry in wavefield time evolution equations If the equations of motion can be written as a set of first-order partial differential equations, a simple substitution of variables usually suffices to obtain skew-symmetry.so, for the equations describing the time evolution of electromagnetic and acoustic wavefields, little needs to be done, since the differential equations are already first order in time.however, in the case of elastodynamics, we must first reduce the equations to a set of first-order equations. 6.. Electrodynamics Recall that without sources and losses, the Maxwell Equations describing the time evolution of the electric field E and the magnetic field H can be written in matrix form as [] H t E = µ ε H E, 6.

3 88 Elastodynamics where ε and µ are resp.the electric permittivity and magnetic permeability.now, we antisymmetrize this equation by substitution of X = µh, Y = εe. 6. The resulting equation is X = µ ε t Y ε µ X Y Acoustics For acoustic waves, again without the presence of sources and losses, the governing matrix equation is [87, 88] p t v = κ x ρ ρ ρ κ y κ z x y z p v. 6.4 Here, ρ is the mass density, κ the compressibility, p the pressure and v = v x, v y, v z T particle velocity.this equation can be anti-symmetrized by substituting the p = p κ, v = v ρ, yielding the skew-symmetric matrix equation t p v = ρ ρ y κ x x ρ z ρ κ y ρ κ κ z ρ κ κ p v 6.5a 6.5b Elastodynamics The linear deformation of a material can be expressed in terms of stretch per unit length or strain.the strain e ij is a symmetric tensor, and to first order equal to a combination of the spatial derivatives of the displacements u i,wherei = x, y, z, e ij = u j + u i. 6.7 x i x j The stress p ij, the force per unit area in a specific direction, is also a symmetric tensor, and according to Hooke s law, it depends linearly on the strain and the elastic properties of the material: p = C e. 6.8

4 6. Skew-symmetry in wavefield time evolution equations 89 where C = C ijkl is the stiffness matrix containing, due to symmetry, at most different components.the total force per unit volume is given by the divergence of the stress F i = p ij, x j j and therefore, the equation of motion can be written as [89 9] ρ u i t = F i = j x j 6.9 { C ijkl u j + } u i, 6. x i x j where ρ is the scalar mass density. In the case of an isotropic elastic solid, the stiffness tensor only contains two different parameters, the Lamé material parameters λ and µ.if we write p = p xx, p yy, p zz, p yz, p zx, p xy T, matrix C for isotropic elastic solids is C = λ + µ λ λ λ λ+ µ λ λ λ λ+ µ µ µ µ 6. In this case, the equations of motion can be written as ρ u i t = λ + λ div u + grad µ grad u i + u + µ u i + µ div u. 6. x i x i x i x i When λ and µ do not depend on space, this equation reduces to [89] ρ u t = λ + µgrad div u + µ u. 6.3 This expression can be separated into two independent equations; one for the movement of individual particles parallel to direction of propagation of wave: compressional or P-wave: ρ u p = λ + µ u t p, 6.4 and one for the movement perpendicular to direction of propagation: shear wave or S-wave ρ u s t = µ u s. 6.5 The most general equation of motion derived so far, i.e.eq.6., is second order in time. A prerequisite of the first step in our scheme to construct new algorithms is that we have a set of equations that is first order in time.a naive approach would be simply defining v = u/t, to obtain a set of first order equations, but the resulting equations are hard to anti-symmetrize.nevertheless, with some caution it is possible to reduce the equations to usable first-order equations.

5 9 Elastodynamics By using the symmetry properties of the stress tensor, Eq.6.9 can be symmetrized to F i = p ij + p ji. 6.6 x j x j j Note that in our description of the state of the system we must include all nine components of the stress tensor, instead of six.the equation of motion can now be written as ρ v t = D p, 6.7 where v is the particle velocity vector and D is the matrix containing the spatial derivatives now in the basis of p = p xx, p xy, p xz, p yx, p yy, p yz, p zx, p zy, p zz T D = x By observing that t e ij = it follows that y z x x y z y x x i v j + z z y x y z. 6.8 v i, 6.9 x j p t = CDT v. 6. Combining Eqs. 6.7 and 6., the resulting first-order matrix equation becomes p t v = CD T ρ D p v. 6. One way to anti-symmetrize this equation, is to focus on the conservation of the total elastic energy.per unit volume, the energy density U is given by the sum of the kinetic and potential energy, U = ρ v + C ijkl e ij e kl. 6. ijkl In analogy with the electromagnetic case, an orthogonal time evolution transformation should conserve the total energy, so, if the state of the system is given by ψt, thenψt should obey Udr = ψt dr 6.3 V V A convenient choice for the components of ψ that fulfills this condition is ψt = w, s T, 6.4

6 6. Skew-symmetry in wavefield time evolution equations 9 where w = ρ v 6.5 and s = p/ C. 6.6 The square root of the stiffness matrix exists because it is symmetric and positive definite. The resulting, skew-symmetric, matrix equation for ψt finally becomes, t ψt = CD T ρ ρ D C ψt Hψt. 6.7 Using the symmetric properties of ρ and C, one can prove that the matrix is skewsymmetric: H T = ρ D C CD T ρ T T = CD T ρ D C ρ = H 6.8 It is instructive to see the explicit form of the time evolution matrix, and for notational convenience we consider the case of a two-dimensional isotropic elastic solid, for which we have where t ψt = α x β ρ x α ρ y β ρ x β ρ y α ρ y ρ x µ µ ρ y ρ x µ ρ ρ x µ y µ y β y ρ α y ρ µ ρ x µ ρ x µ ρ ρ ψt 6.9 α = λ + µ + µ, 6.3 β = λ + µ µ 6.3 and ψt = s xx t, s yy t, s xy t, s yx t, w x t, w y t T. 6.3

7 9 Elastodynamics Figure 6-: Unit cell of the three-dimensional staggered grid onto which the continuous velocity and stress fields of the elastodynamic equations are mapped in order to conserve the skew-symmetry.middle: grid for elastic isotropic solids.note: the Lamé constants λ and µ coincide with the stress field components, and the mass density is only defined on velocity field points.right: grid for the general case i, j = x, y, z. Figure 6-: Unit cell of the two dimensional staggered grid onto which the continuous velocity and stress fields of the elastodynamic equations are mapped in order to conserve the skew-symmetry. 6. Spatial discretization In order to conserve the skew-symmetry for the Maxwell Equations during spatial discretization, the fields are mapped onto the staggered Yee grid and a central point approximation is used for the spatial derivatives that is, f x f x+h f x h/h.for the algorithm solving the elastodynamic equations, a similar, staggered, grid is used [9], a unit cell of which is shown in figure 6-.The spatial derivatives are also approximated by the central point formula, in order to conserve skew-symmetry.the boundary conditions in elastodynamic systems are either free, meaning that the stress field vanishes on the boundary, or rigid, when the velocities are all zero on the boundary.we will not consider absorbing boundary conditions in this thesis. We will now assume that the systems consists of a two dimensional isotropic solid; i.e., all material parameters density and Lamé coefficients, do not depend on the z-coordinate. The spatial grid reduces to the grid shown in figure 6-. The fields are mapped onto this grid, and we adopt the convention that the s xx and s yy stress fields are located in the, corner of the grid.the values of the discretized fields

8 6.7 Conclusion 99 Time-step s Vir VNS- VNS-4 LTS- LTS Table 6-: Error for each of the four algorithms for the time simulation of a random system, as compared with the Chebyshev solution, which uses K = 3 expansion terms. The system measures L x = L y =, withameshofδ =., and the material parameters ρ, λ, µ vary randomly in space with values distributed randomly in the interval [, 3].The error is determined at t = 3.all quantities are expressed here in dimensionless units. 6.7 Conclusion In this chapter, we have shown how the method to construct algorithms to solve the timedependent Maxwell Equations can be applied to other wavefield propagation phenomena. We have implemented four new algorithms to solve the time-dependent elastodynamic equations, and measured the performance to solve some specific seismic problems.the scaling of the error of the Virieux and LTS algorithms compared with the solution generated by the Chebyshev algorithm indicates the correctness of all implementations. The properties of the algorithms are comparable to the properties we found for the algorithms solving the Maxwell Equations.First of all, the accuracy of the solution of the timestepping algorithms depends on the specific initial condition used.for explosive initial conditions, the Virieux type algorithms seem to produce more accurate results than the energy-conserving LTS family algorithms.on the other hand, for random initial conditions, the LTS algorithms are more accurate.if high accuracy is important, the Chebyshev algorithm is the most efficient algorithm to use.

9 6.3 Time integration 93 f are related to their continuous counterparts g by f i, j, t = giδ/, jδ/, t Due to the staggered nature of the grid and the choice of the origin, the s xx and s yy stress fields are only defined on the x =odd and y =odd lattice points.similarly, the w x and w y velocity fields are defined at resp.the x =even/y =odd and x =odd/y =even lattice points, and the s xy and s yx stress fields are given at the x =even and y =even grid entries.we note that all for simplicity of notation the fields are indexed on the full grid, despite the fact that they are not defined on each point. We assume the total number of lattice points in each direction to be odd, and also assume that the boundary is located at the first and last rows/columns of the grid.the boundary conditions themselves are implemented by excluding the field points that are located at the boundary and should be remain zero, from updating during the time integration. Using this grid and the central-difference approximation to the spatial derivative, we obtain the spatially discretized analogue of equation 6.9, for example t s xxi, j, t = αi, j δ w xi +, j, t w xi, j, t ρi +, j ρi, j + βi, j w y i, j +, t δ w yi, j, t, 6.34 ρi, j + ρi, j and t w xi, j, t = αi +, js xxi +, j, t αi, js xx i, j, t δ ρi, j + βi +, js yyi +, j, t βi, js yy i, j, t δ ρi, j + δ [ µi, j + sxy i, j +, t + s yx i, j +, t ρi, j µi, j s xy i, j, t + s yx i, j, t ] Similar equations hold for s yy, s xy, s yx and w y. 6.3 Time integration The formal solution of 6.7 is ψt + τ = expτhψt Due to the skew-symmetry of the matrix H occurring in this equation, the same techniques as employed for the Maxwell Equations can be used to perform the time integration.however, a slight difference in notation is unavoidable, as some fields are defined on identical

10 94 Elastodynamics lattice locations.a third index, specifying the appropriate field, is added to the spatial indices to indicate a field point within the total state vector.therefore, the discrete analogue Ψt of ψt = s xx t, s yy t, s xy t, s yx t, w x t, w y t T is defined by Ψi, j, s xx, t = e T i, j,s xx Ψt = s xx i, j, t, 6.37 and analogous equations apply for indexing the other stress and velocity fields within Ψ. Using this notation, we can write the discretized time evolution matrix H as H = H x,s xx,w x + H x,s yy,w x + H x,s xy,w y + H x,s yx,w y + H y,s xx,w y + H y,s yy,w y + H y,s xy,w x + H y,s yx,w x, 6.38 where for instance the explicit form of H x,s xx,w x is given by H x,s xx,w x = n y n x j= i= n y n x αi +, j δ [ ei, j,sxx ρi, j et i+, j,w x e i+, j,wx e T i, j,s xx] αi, j + j= i= δ [ ei, j,sxx ρi +, j et i+, j,w x e i+, j,wx e T i, j,s xx] 6.39 = H x,s xx,w x + H x,s xx,w x. 6.4 Here, the prime in the summation indicates that the summation index is increased with strides of two.it is easy to convince oneself that the matrices H x,s xx,w x and H x,s xx,w x are block diagonal and skew-symmetric.the other matrices in equation 6.38 have similar explicit forms and can also be decomposed into block diagonal parts.therefore, a firstorder approximation to the matrix exponent occurring in equation 6.36 is expτh = exp τh x,s xx,w x exp τh x,s xx,w x x,s exp τh yy,w x x,s exp τh yy,w x exp τh x,s xy,w y x,s exp τh xy,w y x,s exp τh yx,w y x,s exp τh yx,w x exp τh y,s xx,w y y,s exp τh xx,w y y,s exp τh yy,w y y,s exp τh yy,w y exp τh y,s xy,w x y,s exp τh xy,w x y,s exp τh yx,w x y,s exp τh yx,w x +Oτ. 6.4 The computation of each of these matrix exponents only involves repeatedly rotating two elements within Ψ, since the matrix exponent of these block diagonal matrices is the matrix composed of the exponents of the skew-symmetric blocks themselves, which is a rotation. 6.4 Implementation We now implement four different kinds of algorithms to simulate the time evolution of elastodynamic wavefields:

11 6.5 Sources 95.The conventional algorithm as developed by Virieux [9], which can be regarded as a seismic FDTD implementation: the fields are defined staggered in time and defined on the staggered grid as shown in figure 6-.The spatial derivatives are approximated by using central differences.this algorithm will be referred to as Vir and is second-order accurate in time..the staggered-in-time nature of the Vir algorithm can be removed by a procedure completely similar to one used for the time dependent Maxwell Equations, as explained in section.3.. The algorithm obtained in this way will be referred to as VNS- or VNS-4, depending on the order of accuracy in time. 3.The algorithm obtained by using equation 6.4 to approximate the matrix exponent. This is the equivalent of the Lie-Trotter-Suzuki algorithm developed for solving the time dependent Maxwell Equations, hence we will refer to it as LTS- or LTS-4, depending on the order of accuracy in time. 4.An algorithm based on approximating the matrix exponent of the matrix in equation 6.38 using a Chebyshev expansion. It will be referred to as Chebyshev. 6.5 Sources The most natural initial condition to use to simulate seismic phenomena is an explosive source [9].In our case, we assume the source to excite the s xx and s yy fields at a specific point, and with time dependency f t = exp αt t, 6.4 or its derivative gt = t f t = αt t exp αt t The parameters t and α are chosen such that the contribution of the source is negligible at t =. For the timestepping algorithms, the implementation of a source is trivial and explained in section.4. However, for the Chebyshev algorithm, some work needs to be done to expand the source term in Chebyshev polynomials. We proceed as in section.4.. In the presence of source f t with spatial extension sr, the solution of the time evolution of the fields becomes Ψt = e th Ψ + t due t uh f usr = e th Ψ + ht, Hsr Some caution needs to be taken since the eigenvalues of H are all imaginary.for a function hix, withx [, ], the Chebyshev expansion is given by hix = a n i n i n T n ix = n= b n T n ix 6.45 n=

12 96 Elastodynamics where T n is the modified Chebyshev polynomial defined in section.., and the prime in the summation indicates that the first term in the summation is divided by two.the expansion coefficients are given by π b n = i n a n = i n dθ hi cos θ cos nθ π So, we need to compute these expansion coefficients for the source term ht, H as function of H.We start by evaluating ht, H = ht, t,α,h for the Gaussian source 6.4: ht, t,α,h = t [ erf due t uh e αu t = π α exp t t H + H /4α ] t α + erf H α t t α + H α 6.47 In order to guarantee proper arguments for function ht, H = ht, cos θ in equation 6.46, we normalize matrix H by substitution of z = H, z = t H, β = α/ H and x = ih/ H, obtaining π hz, z,β,x = [ erf α exp z z ix x /4β ix z β β + erf z z β + ix ] β The value of the complex valued error functions may become large for large fractions H /α as x, but this effect is cancelled by the presence of the factor exp x /β.now we put x = cos θ and perform the remaining integral over θ in equation 6.46 by a Fast Fourier transformation: N b n = i n e πink/n hz, z,β,cos πk N k= Here, we made use of the fact that for any complex function f for which R f x = R f x and I f x = I f x, and this holds for hz, z,β,x, we have π π e inθ π cos nθ R f cos θ, if n is even f cos θdθ = π 6.5 i π π cos nθ I f cos θ, if n is odd Therefore, the expansion coefficients are all real. The derivation of the Chebyshev expansion coefficients for a source defined by equation 6.43 is very similar, and will not be treated explicitly here. 6.6 Results Consider a rectangular system consisting of two different materials, displayed in figure 6-3. This system is also studied in references [9,93], and proved to be a good testing situation for the performance of an algorithm solving the elastodynamic equations.we simulate the

13 6.6 Results 97 Figure 6-3: The corner-edge system, consisting of two different materials.the system size and location of the source are indicated in the picture.at the top, a free-surface boundary condition is imposed, the other boundaries are rigid.the overall density is ρ = 5 kgm 3, and the mesh size is δ = m.in the bulk material I, we have v p = 6 kms and v s = kms, whereas in the inner material II, the wave velocities are v p = 9 kms and v s = 3 kms.thesourcex excitesthes xx and s yy stress fields with time dependence gt from equation 6.43 and parameters α = 4 and t =.5 s. Figure 6-4: Two snapshots of the kinetic-energy density distribution of the corner-edge system of figure 6-3.Left: state at t =.8 s, right: state at t = 6. s. time evolution of the velocity and stress fields after an explosion modeled by equation 6.43, up to t = 6 s, with the four different algorithms.in figure 6-4, we show the kinetic-energy density distribution at two different time instances. To analyze the efficiency of each algorithm in solving the time evolution, we follow the same procedure as for the Maxwell Equations, see section 3.. We compare the error, defined as Ψ Ψ / Ψ + Ψ, with the solution obtained by using the Chebyshev algorithm.for the corner-edge system, the errors are listed in table 6-.We see that for the largest timestep, the Virieux type algorithms Vir,VNS-,VNS-4 are unstable.this can be expected, since the maximum timestep is limited by the largest velocity, the v p velocity, and the mesh-size, through the Courant limit [9] τ< δ v p. 6.5 Furthermore, we see from table 6- that for all algorithms the error scales according to the order of accuracy in time, similarly to the results we found for the Maxwell Equations, in

14 98 Elastodynamics Time-step s Vir VNS- VNS-4 LTS- LTS Table 6-: Error for each of the four algorithms for solving the time evolution in the corneredge system of figure 6-3, as compared with the Chebyshev solution, which uses K = 54 expansion terms.a dash - indicates that the error was not measured for this parameter, whereas an infinite symbol denotes that the algorithm was not stable.the error in the staggered-in-time Vir algorithm is determined by averaging the error in the kinetic and potential energy density at respectively the final time instance and the final time instance shifted by half a timestep.the same time shifting procedure is carried out by the Chebyshev algorithm and also applied to prepare the initial condition. section 3.. It is clear that for the corner-edge system, the Virieux type algorithms perform much better than the energy-conserving LTS type algorithms, as long as the timestep is smaller than the Courant limit.for timesteps larger than the Courant limit, the LTS family algorithms are stable, but the error does not yet scale according to the order of accuracy in time. With respect to the efficiency of the algorithms, we note that the number of matrixvector operations W, necessary to perform one timestep, is for the Vir algorithm,.5 for the second-order VNS- and LTS- algorithms, and for the fourth-order VNS-4 and LTS-4 algorithms.for the specific example here, the corner-edge system, the Chebyshev algorithm employs K = 54 expansion terms.at t = 6 and τ =.5, the VNS- algorithm already uses more namely 8 matrix-vector operations.therefore, we draw the conclusion that Chebyshev algorithm should be preferred to be used to solve the time evolution for this problem.note that in general, the choice of which algorithm to use depends heavily on which degree of error is acceptable.in this specific case, there are values for τ for which the VNS- algorithm uses less matrix-vector operations than the Chebyshev algorithm, but then the error will be larger than the error for τ =.5, and maybe unacceptably high.on the other hand, we emphasize that in practice, one VNS- or LTS- matrix-vector operation is carried out faster than one Chebyshev recursion iteration, although this depends one the actual implementation. It is important to note that the initial condition plays an important role in the error of the solution produced by a specific algorithm.from the results of the corner-edge system, one might draw the conclusion that the Virieux type algorithms achieve better results than the LTS family for all systems.this is not true.in table 6-, we list the error as function of timestep for all algorithms, as compared with the Chebyshev algorithm, for a system consisting of a random medium and starting from a random initial condition.from the table, it is clear that in this case, the LTS family of algorithms perform better than the Virieux type algorithms.

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