AN EXPLICIT METHOD FOR NUMERICAL SIMULATION OF WAVE EQUATIONS

Transcription

1 The 4 th World Conference on Earthquake Engineering October -7, 8, Beiing, China AN EXPLICIT ETHOD FOR NUERICAL SIULATION OF WAVE EQUATIONS Liu Heng and Liao Zheneng Doctoral Candidate, Det. of Structural Engineering, Institute of Engineering echanics, Harbin,China Professor, Det. of Shibuilding and Ocean Engineering, University of Engineering, Harbin. China Eail: liuheng@ie.net.cn, liaoz@hrbeu.edu.cn ABSTRACT: A ethod to develo a hierarchy of exlicit recursion forulas for nuerical siulation in irregular grids for scalar wave equations is resented and then is used to construct the forulas for the one-diensional case in this aer. Nuerical siulation of the one-diensional scalar wave equation in a regular grid is discussed for understanding its accuracy and stability, and an aroach is then roosed to construct the stable forulas which are of -order of accuracy both in tie and sace with being a ositive integer and the recursion forulas of the second order (=) and the fourth order (=) are given as an exale. Theoretical results of the ethod are deonstrated by a series of nuerical tests. KEYWORDS: wave equation, nuerical siulation, exlicit recursion forula, finite eleent ethod (FE). RECURSION FORULAS AND SOLUTIONS OF THE INITIAL-VALUE PROBLE Our starting oint is a concet that wave seed is finite. According to this concet we will first clarify the relationshi between the exact solution of an interior oint of a finite hoogeneous area within a short tie window and solution of the initial-value roble for an infinite hoogeneous sace in the section, and the recursion forulas for an irregular grid are then derived by interolation aroxiation. Assuing the wave seed is a constant in a finite satial doain, the field of dislaceent and that of velocity satisfying a scalar wave equation are denoted by uxt (, ) and vxt (, ) resectively, where x denotes a coordinate vector, and t eans tie. If the dislaceent distribution function ux (,) and the velocity distribution function vx (,) are known, we investigate u(, t) and v(, t) at a oint P which is assued to be located at the coordinate origin without losing generality. We assue that the shortest distance fro P to the boundary of the area is l, the neighborhood of P is defined as x x, < x l. Let t x c,the following udgents can be drawn fro the concet of finiteness of wave seed: u (, t) and v(, t) are deterined coletely by ux (,) and vx (,) on the interval x x, and have nothing to do with the otion of all the other oints outside the neighborhood of P as t = for their effects have not reached the oint P at t = t. Therefore, we can extend the neighborhood of P with a constant seed to an infinite hoogeneous sace as far as couting the otion of P at an adacent next tie is concerned. So, it ilies the otion of P can be couted using the solution of Cauchy roble and the for of the couting forulas are as follows: ( ) = u ( ( x ) ( x ) ) ( ) = v ( ( x ) ( x ) ) u, t J u,,v, v, t J u,,v, (.) Where the function u ( ( ) ( ) ) u (, x ) and v (,) x are defined on the interval x c t, and the exressions of functional J u x,,v x, can be derived fro solutions of the Cauchy roble. The solutions are well known for

2 The 4 th World Conference on Earthquake Engineering October -7, 8, Beiing, China the wave equations in -D, -D and 3-D cases, which are called as D Alebert, Poisson and Kirchhoff forula resectively [] ; and ( ) v J.,. can be derived fro J ( ) u.,. by differentiation with resect to tie. As far as the nuerical si ulation of wave equation i n tie doain is concerned, a continuous sace is required to be discretized via a grid and to construct the recursion forulas of the nodal oints. If data of the otion are known at the oint P and the adacent nodal oints when t =, the distribution function u ( x,) and v ( x,) in the neighborhood of P ( x c t) can be aroxiat ed in ters of these discrete data via inter olation. Substituting these aroxiate distribution functions into Eqn.., a basic for of the recursion forulas can be obtained.. THE BASIC RECURSION FORULA FOR THE -D CASE For the nuerical siulation of the -D wave equation, the continuous x axis is discretized by a sequence of satial discrete oints P with coordinate x = x, =, ±, ±,. Point P and the adacent nodal oints P, =±,, ± with being a ositive integer consist of a lo cal syste of nodal oints and Fig.shows the case of =. x x x x x x Introducing the discrete tie t Figure Scheatic for a local syste of nodal oints in a -D irregular grid = t with being an integer nuber, we define that (, ), (, ) u = u x t v = tv x t (.) Alying the aforesaid ethod to the -D wave equation, the basic for of recursion forulas of a nodal oint [] for a -D irregular grid is obtained + = + + ξi + i + ηi +i i= i= u u v u v + = + ζi + i + γi + i i= i= v v u v (.) If the grid is regular, then x = x, so Eqn.. can be reduced to + u = u + v + ( ξ ( u u + u+ ) + η( v v + v+ ) ) = ( ζ ( + ) γ ( + )) v = v + u u + u + v v + v + = (.3) After analyzing accuracy and Von - Neuann stability [3] of Eqn..3, we found that the forulas are of + -order of accuracy but unstable when coefficients ξ, η, γ and ζ, resectively, satisfy the

3 The 4 th World Conference on Earthquake Engineering October -7, 8, Beiing, China following four systes of equation τ = = l dlξ τ, dlη = = l ( l + ) l l dlζ = l τ, dlγ = τ = = l =,, (.4) l Where dl =, τ = c t/ x. And we also found that the two systes of equation about coefficients ξ and η guarantee the error ters of even and odd order in eu (truncation error of u ) equal to zero u to the order of + resectively; so does the grou of coefficients ζ or γ for ev (truncation error of v ); each syste of equation contains equations nubered as l =,,, and each equation in the syste only ensures the error ter of the corresonding order equal to zero in the truncation error. For exale, the l th equation in the syste about η ensures the error ter of the ( l + )th order in eu equal to zero, while is not related to the other lower-order ters; articularly, the last equation with the sequ ence nuber l = only ensures the ( + )-order ter in eu equal to zero, but has nothing to do with other error ters of order lower than +. Therefore, Eqn..4 akes it ossible to adust the coefficients of the recursion forulas while satisfying requireent to the lower order of accuracy. Thus if the last equation in the syste about η is eliinated, that is, if we abandon requireent of ( + )th order error ter in eu equal to zero, any value can be assigned to one of η, η.so, we roose an aroach to construct stable recursion forulas of -order in the next section. 3. AN APPROACH TO DEVELOP STABLE RECURSION FORULAS OF -ORDER FOR THE REGULAR GRID The order of accuracy of Eqn..3 deends on values of the coefficientsξ, η, ζ andγ, whether Eqn..3 satisfies the Von Neuann stability condition deends on the values as well. Consequently, it is ossible to develo stable recursion forulas of -order of accuracy by selecting values of ξ, η, ζ andγ reasonably. There are several aroaches to adust the coefficients, and the one we roose is as following: values of ξ, ζandγ ( =,, ) are still solved by Eqn..4; new values η are assigned to η ( =,, ). Substituting η = η into the second syste of equations in Eqn..4 and abandoning the last o ne lead to l τ dlη = + η, l,, = = (3.) ( l + ) The new values of η η can be solved fro Eqn.3. as functions of η. Let η = η, where η is also solved by Eqn..4, and is an adustable araeter. If the value of η selected satisfies thevon Neuann stability condition via adusting, Eqn..3 will be stable and of the -order of accuracy as η in Eqn..3 is relaced by η. Thus our roble now becoes to search values of which satisfy thevon Neuann stability condition for a given τ (< τ ). A region coosed of all such values of is called as the stability region of, the one in the region closest to is denoted as, which is called the otial value for the iniu loss of accuracy as =.Substituting η = ηinto Eqn.3.,we can obtain the otial coefficients η which ake Eqn..3 stable and of iniu accuracy loss with ξ, ζandγ ( =,, ) still solved by Eqn..4. As an

4 The 4 th World Conference on Earthquake Engineering October -7, 8, Beiing, China exale, the coefficients and the stability region of fo r the stabl e recursion forulas of the second order ( = ) and the fourth order ( = ) are given as follows: = : τ τ τ ξ =, η = η =, ζ = τ, γ = (3.) 6 where 5. = : = for all (,] τ,and the stability region of shrinks to be a oint = = τ τ τ τ τ η = η =, η = η = τ τ τ τ ξ = γ = +, ξ = γ =, τ 4 τ τ τ ζ = +, ζ = (3.3) τ Table Part of the nuerical results of versus τ /34 3 stability region 45/ τ Figure The stability region of and versus. Figure shows the stability region of (the shadow art) searched nuerically within the range,3. τ [ ] 4. Nuerical tests Con sidering a Cauchy roble for the -D wave equation with wave seed c= 5 / s, vx (,) =, ux (,) is a cubic B sline function (See Eqn. 35, P8 in [4]), whose non-zero art are lo cated on the in terval x [,5]. A series of nuerical tests for different values of τ and x are carried out to verify the results resented in this aer, and the ain results are as follows:

5 The 4 th World Conference on Earthquake Engineering October -7, 8, Beiing, China 4.. Stability Verification The nuerical results indicate that Eqn..3 with a value of in the stability region is stable, otherwise it s not. For exale, Figure 3 shows the tie histories of dislaceent ( u ) and velocity ( v / t) at x = for the with τ =, x =.5, t =.s. The value of equals 4 and for the stable recursion for ulas of the second order and that of the fourth order, resectively, while = for the unstable forula. Figure 3 indicates clearly that the instability henoena occur quickly in the nuerical results for the unstable schee; eanwhile, the nuerical results for the stable forula atch the exact solution very well. initial eriod t [,.6s] dislaceent. exact stable unstable velocity. exact stable unstable tie/s (a) =:dislacent history tie/s (b) =:velocity history..5 exact stable unstable..5 exact stable unstable dislaceent. -.5 velocity tie/s tie/s (c) =:dislaceent history (d) =:velocity history Figure 3 Coarison between the stable and the unstable recursion forulas 4.. Accuracy Verification Let ux (,) be a cubic B sine function and τ =, x =.5, t =.s. Figure 4 shows the wavefors of dislaceent ( u ) and velocity ( v / t) at t =. 5sand at this tie the initial disturbance has roagated 5 to the left and the right resectively. It can be seen fro the figure that the nuerical solutions of the stable recursion forulas of the second and fourth order are very close to the exact solution. In order to observe effects of the accuracy order on errors of the nuerical solutions, the error function of dislaceent δu(,, τ ) and that of velocity δ v(,, τ ) are defined by δu(,, τ) = u u( x, t) δv(,, τ) = v tv( x, t), J, (4. )

6 The 4 th World Conference on Earthquake Engineering October -7, 8, Beiing, China The error nor of dislaceent δu and that of velocity δ v are introduced to evaluate the nuerical accuracy of the whole wavefor at t = t, δu = ( δu(,, τ)) δv = J = J = ( δv(,, τ)) (4.) Where the value of J should cover all nodal oints whose otion data was used by Eqn..3 till the given tie t. The error nors are couted by Eqn.4. for each value of τ for τ =. n, n =,,.Figure 5 show s the change of δ u and δ v versus τ at t = 5. s. It can be seen fro it that the error nor of the fourth order forula is considerably saller th second order one for alost the entire interval τ,. than at of the ( ] dislaceent exact second order fourth order velocity exact second order fourth order x/ (a) dislaceent wave for x/ (b) velocity wave for Figure 4 The wavefor of dislaceent and velocity at t = 5. s. second order fourth order A.3 second order fourth order. δ u. δ v τ A Figure 5 The error nor of dislaceent δ u and the error nor of velocity δ v τ

7 The 4 th World Conference on Earthquake Engineering October -7, 8, Beiing, China Finally we design three schees with τ =.5to reveal the relationshi between the order of accuracy and the coutational efficiency: () the second order stable forula with x =.5 and t =.5s; () the fourth order stable forula with x =. and t =.s ; (3) the second order stable forula with x =. and t =.s.figu re 6 shows the error δ u(,,.5) and δ v(,,.5) / t,where the two grou s of data have been nor alized by the axiu of the exact solution of dislaceent and velocity, resectively. It shows that error of the schee () shares the sae order of agnitude as that of the schee (), and both of the are considerably saller than error of the schee (3). This indicates that the fourth order schee has alost the sae nuerical accuracy as the second order stable one though the tie ste and sace ste of the forer is one tie larger than those of the latter. It eans that the iroveent of the order of accuracy can not only ake the nuerical siulation ore accurate, but also irove coutation efficiency for sae recision... schee () schee () schee (3).5 schee () schee () schee (3) δ u. δv/ t tie/s Figure 6 Coarison between the recursion forulas with different order of accuracy and different gird size τ =.5 ( ) tie/s 5. CONCLUSION The aer resents a ethod to develo the recursion forula which is exlicit, tie-sace decouling, highly accurate, stable and single-ste for nuerical siulation of wave equation in irregular grids and in tie doain, according to the concet of wave seed being finite; and deonstrates the feasibility for the ethod via -D odel. Having noticed the well-known solutions for the -D and 3-D Cauchy roble, the ethod can be alied to constructing the recursion forula for ultidiensional irregular grids. We have generalized this ethod to the -D odel, and the generalization to the 3-D case is in rogress. And the ethod is suitable for the nuerical siulation in a sace doain where wave seed varies soothly with satial coordinates as long as the wave seed in the recursion forula takes a value of that at a nodal oint under consideration; furtherore, the ethod can be alied to develoing the recursion forula of nodal oints on an interface where an abrut change in wave seed occurs if the solution of initial-value robles for the hoogeneous infinite satial doain is extended to that for an infinite satial doain with the abrut interface. Besides, the ideas and the aroaches resented in the aer have certain reference to iroving techniques for the nuerical siulation of electroagnetic and elastic wave equations. The above generalization is ossible as shown by soe work we have done, the related research results will be resented in forthcoing aers. REFERENCES [] Karl F. G.(975). Wave otion in Elastic Solids, Oxford University Press, U.K. [] Liao, Z.P., Liu, H. and Xie, Z.N.(8).An exlicit ethod for nuerical siulation of wave equations -D wave otion. Chinese Journal of Theoretical and Alied echanics (To aear in Chinese).

8 The 4 th World Conference on Earthquake Engineering October -7, 8, Beiing, China [3] Joe D. H.(). Nuerical ethods for Engineers and Scientists (Second Edition),arcel Dekker, Inc., New York. [4] L iao Z.P.(4). Introduction to Wave otion Theories for Engineering(Second Edition),Science Press, Beiing(in Chinese).