'DO NOT RETURN TO LIB R A R /
EVALUATION AND ANALYSES OF SOME FINITE ELEMENT AND FINITE DIFFERENCE PROCEDURES FOR TIME-DEPENDENT PROBLEMS
by
C. S. Desai, J. T. Oden, L. D. Johnson
Soils and Pavements Laboratory U. S. Army Engineer Waterways Experiment Station P. O. Box 631, Vicksburg, Miss. 39180
April 1975 Final Report
Approved For Public Release; Distribution Unlimited
TA Prepared for Office, Chief of Engineers, U. S. Army 7 Washington, D. C. 2 0 3 14 .W34m S-75-7 1975 PROPERTY OF SUREAU OF RECLAMATION BUAEA.y..?,ÌlECLAMAT,0N DENVER LIBRARY -TA 92031039 Unclassified ^ 5 ^ 0 , \ SECURITY CLASSIFICATION OF THIS PAGE (When Data Enter'd) s -7^7 BEFORE COMPLETING FORM 2. GOVT ACCESSION NO. 3. RECIPIENT’S CATALOG NUMBER h-?v 1. R E P O R T N U M B ER Miscellaneous Paper S-75-7 5. TYPE OF REPORT & PERIOD COVERED 4. T IT L E (end Subtitle) Final report EVALUATION AND ANALYSES OF SOME F] [NITE ELEMENT AND FINITE DIFFERENCE PROCEDURES I PO R T I M E - 6. PERFORMING ORG. REPORT NUMBER DEPENDENT PROBLEMS
8. CONTRACT OR GRANT NUMBERfe) 7. AUTHOR^; C. S. Desai J. T. Oden’
L . D . Johnson 10. PROGRAM ELEMENT, PROJECT, TASK 9. PERFORMING ORGANIZATION NAME AND AUU k ESS . AREA & WORK UNIT NUMBERS U. S. Army Engineer Waterways Experiment Station Soils and Pavements Laboratory P. 0. Box 631, Vicksburg, Miss. 39180 12. R E P O R T D A T E 11. CONTROLLING OFFICE NAME AND ADDRESS Office, Chief of Engineers, U . S. Army April 1975 Washington, D, C. 2031^- 13. N U M BER O F PA G ES 107 15. S E C U R IT Y CLASS, (of this report) 14. M O N ITO R IN G AG EN C Y NA M E & ADDRESS(7f different from Controlling Office) Unclassified
15a DECLASSIFICATION/DOWNGRADING SCHEDULE
16. D IS T R IB U T IO N S T A T E M E N T (of thia Report) Approved for public release; distribution unlimited.
17. D IS T R IB U T IO N S T A TE M E N T' (at the abstract entered ,n Block 20, It dltterent Iron, Report)
18. SUPPLEMENTARY NOTES
19. K E Y W O RDS (Continue on reverse side if necessary and identify by block number) Finite element method Finite difference method Time dependence Soil mechanics
------” . , r t ______nrrsst } riser, Hfv hv h lo c k num ber) A large number of scnemes using une i u u w --- - o (FD) methods have been proposed. Numerical and physical properties of these schemes, vital for their use as general procedures and significant from the viewpoint of the user, are often not examined adequately. One of the aims f this study was to examine numerical properties of a selected^number of schemes for problems relevant to geotechnical engineering. In this initial phase, the aim has been partially fulfilled by studying the problem of one-dimensional
EDITION OF t NOV 65 IS OBSOLETE DD 1473 Unclassified ______SECURITY CLASSIFICATION OF THIS PAGE < ™ « > uera Entered) Unclassified. SECURITY CLASSIFICATION OF THIS P»GE(TW.«. D.lm Bnt.fdi
20. ABSTRACT (Continued).
consolidation and wave propagation. Another aim of the research was to examine the concept of space-time finite elements for some time-dependent problems. A portion of Section I involves quantitative analyses of such factors as con- sta^ i t y , and computational efforts for a number of FD and FE scheme* for the consolidation problem governed by a linear parabolic equation Also examined are the effects of the order of the approximation models and heavier i ^ s o l u t i o n at different locations in space and in time. Mathematical stability criteria are also derived for two time integration schemes with two approximation models for the linear parabolic equation. In Section II, Prof. . T. Oden has presented mathematical derivations for error estimates of FE ap proximations to the diffusion equation. The numerical characteristics have been examined and error estimates have been obtained by using semigroup theo retic, energy, and L? methods. The work presented in this section is father fundamental in nature~and can provide basis for needed investigations towards numerical analysis of FE procedures. Part V of Section I contains formulations of the concept of space-time finite elements for dynamical systems, and for owo equations of fluid flow through porous media. A simple example of the srngle-degree-of-freedom system is solved quantitatively. Since the proiect had to be postponed, no quantitative analyses and definitive conclusions were derived for the concept of space-time elements. The topic of evaluation of different numerical schemes is of vital importance. This is particularly so because a large number of schemes have often been proposed without adequate guidelines for the user to assist hit, in selectinglhe o p t i ™ scheme his specific needs. It is felt that the work initiated in this study should be analysed^and^ev^!uatedf0r <* P ™ « - ! ¡«Prance he
- ______Unclassified______SECURITY CLASSIFICATION OF THIS PAGE(Tf7ien Data Entered) PREFACE
The study of the concept of finite elements in the time domain and evaluation of some finite element (FE) and finite difference (FD) schemes was initiated by the Soils and Pavements Laboratory, U. S. Army Engineer Waterways Experiment Station (WES), Vicksburg, Mississippi. This work was sponsored by the Research in Military Engineering and Con struction (RMEC) group of the Office, Chief of Engineers, U. S. Army. The investigations described in Section I were performed by Dr. C. S. Desai, Research Group, Soil Mechanics Division, now with the Department of Civil Engineering, Virginia Polytechnic Institute and State University, Blacksburg, Va. Prof. J. T. Oden, consultant to the project, University of Texas, Austin, Tex., contributed Section II, and provided valuable advice and suggestions. Dr. L. D. Johnson, Research Group, Soil Mechanics Division, assisted in computational work related to the analyses of one-dimensional consolidation problems. Prof. R. L. Lytton, Texas A&M University, collaborated on the derivation pre sented in Part IV. Section I of this report was prepared by Dr. C. S. Desai and Section II was prepared by Prof. J. T. Oden. The project was postponed after completion of the initial phase described herein. During the investigation, Mr. J. P. Sale was the Chief, Soils and Pavements Laboratory, and Mr. C. L. McAnear was Chief, Soil Mechan ics Divison. Directors of the WES during the investigation were BG E. D. Peixotto, CE, and COL G. H. Hilt, CE. Mr. F. R. Brown was Technical Director.
iii
CONTENTS Page iii PREFACE ...... 1 SECTION I ...... • • PART I: INTRODUCTION...... 3 PART II: EFFECT OF ORDER OF APPROXIMATION MODELS ...... 5 Finite Element Formulation...... 5 Numerical Characteristics and Comparisons ...... 13 Applications...... 22 Comments...... 31 PART III: EVALUATION OF FINITE ELEMENT AND FINITE DIFFERENCE SCHEMES...... 32 Finite Difference Schemes ...... 32 Finite Element Method ...... 33 Comparisons ...... 33 Illustrative Example...... 1+0 Comments...... 1+1 PART IV: DERIVATION OF STABILITY CRITERIA FOR TWO TIME INTEGRATION SCHEMES FOR LINEAR PARABOLIC EQUATION .... 1+3
Linear Model...... 1+3 1+ 1+ Cubic M o d e l ...... Discussion...... 1+7 PART V: FINITE ELEMENTS IN THE TIME DOMAIN ...... 1+8 Dynamical System...... 1+9 Application ...... 52 Fluid Flow Problems ...... 51+ 59 SECTION I I ...... PART VI: SOME ASPECTS OF THE THEORY OF FINITE ELEMENT APPROXIMATIONS OF THE DIFFUSION EQUATION...... 61 6l Introduction...... • • ...... Finite Element Approximation and Interpolation. . .^...... 6 3 Finite Element Approximation of the Diffusion Equation. . . . 68 Error Estimates for the Diffusion Equation Using Semigroup Theoretic Results ...... 70
v Error Estimates for the Diffusion Equation Using the Energy Method ...... 83 Error Estimates for the Diffusion Equation Using L2 M e t h o d s ...... 87 PART VII: CONCLUSIONS AND RECOMMENDATIONS ...... 97 REFERENCES ...... 98 TABLES.1-5
vi SECTION I
by
C. S. Desai, L. D. Johnson
EVALUATION AND ANALYSES OF SOME FINITE ELEMENT AND FINITE DIFFERENCE PROCEDURES FOR TIME-DEPENDENT PROBLEMS
PART I: INTRODUCTION
1. A large number of finite element (FE) and finite difference (FD) formulations has been proposed for solution of engineering problems The question facing the engineer who intends to use these methods is which procedure will provide sufficient accuracy most economically. A number of factors, such as numerical characteristics of the procedures, discretization schemes, physical requirements, geometry and properties of the media, order of approximation, and relative cost can influence selection of the proper scheme. The question has not been answered ade- 1 8 quately; only a few studies have been reported. 2. As a step toward the foregoing aim, eight FD and FE schemes were examined quantitatively from the viewpoints of convergence, sta bility, and computational times. The example of one-dimensional consoli dation governed by a parabolic equation was considered. Effects of such factors as spatial and temporal locations on the numerical solutions and of approximating models were also studied. 3. Often the user tends to establish the reliability of a numeri cal procedure by solving a number of example problems. This pragmatic approach is often necessary; however, it may not necessarily yield a general solution scheme. This is because a procedure may provide satis factory solutions for a subset of problems within the class of problems for which it is devised, and it may not provide consistent answers for another subset. One of the ways to establish generality of a numerical procedure is to examine its mathematical properties, such as convergence stability, and consistency. Hence, in addition to the quantitative study, mathematical expressions for numerical stability were derived for two FE schemes for the linear parabolic equation (Section I, Part IV), and for nonlinear hyperbolic equations, Section II. U. The concept of finite elements in the time domain was
3 investigated. A formulation was obtained for the general dynamical sys tem, and was applied in solving a single-degree-of-freedom system. Formulations based on this concept were derived for a parabolic, hyper bolic, and nonlinear fluid flow equation by using different interpola tion functions. ,5« The study involves rather a small step related to the general and significant problem of selection of the optimum numerical procedure. In the preliminary analysis, only problems governed by one-dimensional equations were considered. Further investigations will be required to answer in detail additional and more complicated aspects of multidimen sional problems. At this stage of the development of numerical methods, the analyst should devote attention to this question and evolve criteria to assist the user in selecting the optimum scheme for his specific needs.
u PART II: EFFECT OF ORDER OF APPROXIMATION MODELS
6. Some alternatives available for improving a FE solution are to use a finer mesh layout with a simple lower order model (or function) or to use a coarser mesh with higher order functions. For a given mesh, a higher-order model will usually yield a more accurate solution. However, a higher order model will give a greater degree of connectivity, thus increasing the band width of the resulting equation set.1, ’ ’ Because the number of operations, say in the commonly used Gauss-Doolittle pro- cedure, is proportional to NB , where N = number of equations and B = semiband width , the equations for the higher order model will take more computation time. Moreover, although the higher order assemblage may involve a smaller number of elements for a given accuracy, the com putation time taken in formulating the element matrices and forcing pa rameter vectors will be greater for the higher order models.
7 . There exist tradeoffs in computational time and cost as well as accuracy that must be examined if the above question is to be answered adequately. In the final analysis, what should govern the selection of an approach is the degree of accuracy per unit cost. This cost should include the expense of preparing the input, the charges for digital l I4. 8 computation time, and the amortization of software development. ’ 5 8. Some of the above aspects were evaluated by adopting two ap proximating models, linear and cubic, for pore water pressure in the parabolic equation governing one-dimensional consolidation^ (or heat flow). The approximation of one-dimensional consolidation can provide useful engineering solutions for many practical problems such as esti mation of vertical settlements of foundations and embankments.
Finite Element Formulation
Background 9. On the basis of Darcy's law, satisfaction of the condition of continuity, complete saturation of porous soils and linear elastic be havior of the soil skeleton, the governing equation for one-dimensional
5 (vertical) consolidation proposed by Terzaghi is
92 u _ 9u
Cv 8x „ 2 " 9t (i) where u = pore water pressure, x = space coordinate, and t = time co ordinate. The coefficient of vertical consolidation, c^ , is expressed as
k (l + e v c = ------in which k^ = coefficient of vertical permeability, e = void ratio of soil defined as the ratio between volume of voids and volume of solids, = unit weight of water, and a^ = coefficient of compressibility of soil. Terzaghi assumed that the total constant vertical stress, a , applied to the soil is carried jointly by the soil skelton and the water in the pores. Accordingly,
a = o' + u (3) in which o' = effective stress in soil and u = pore water pressure. Initially, the total stress a is carried wholly by the pore fluid, i.e. at t=0, o' = 0 , and a = u . As time elapses, the stress is transferred to the soil skeleton and finally at t = 00 , o' = a , and u = 0 . 10. The soil is assumed to deform as a linearly elastic material, and hence, the stress-strain law can be expressed as
da = -■ de a («0 v in which a^ equals the slope of the stress versus void ratio curve. Because of the relationships in Equations 3 and U, it is necessary to ob tain solutions for pore water pressure u only. Hence, solution of Equa tion 1 would also permit computations of deformations in the soil mass.
6 Derivation of finite element equations
11. The following formulation is based on a variational principle for initial value problems by Gurtin,^ and Sandhu and Pister.1^ The boundary conditions considered herein are (Fig. l):
u(0,T ) = U-^x) xe(t - At,t) (5) u U, t ) = u2 (x) xs(t - At,t)
The initial conditions are
u(x,0) = u q (x ) 0 £ x £ l (6 )
Integration of equation 1 over a short time interval, At , gives
t 2 c dx = u(t) - u(t - At) (7) V 3x2 t-Atl which can be rewritten as
~2 I * o U (8 ) 8 cv 7 T = u “ uo dx
where the notation * denotes convolution product and g' * u denotes ,t I udx . The boundary conditions are transformed to t-At
g f * u(o,x) = g' * tLj^x) (9 ) g ! * u(&,x) = g' * u2(x)
12. The governing functional corresponding to Equation 1 can be
written as
32u = J ju * u - U * g' * cv — — - 2u * u q dx (1 0 ) dx
Introduction of nonhomogeneous boundary conditions gives
7 N O . O F MESH N O . NODES
1 6 2 11 3 21 4 26 A<7 A ct
i U(0, T) e Û, (T) r 0 U(0, T) - U^T) = 0 -DRAINAGE ©
U(X,0) = UQ (X) = A a T T © U (X , 0 ) = U q (X ) = A ( 7
ELEMENTS-*
tTm e ~m INTERFACE m u + & + NODES-* m m
u(1, t) = u2(r) = 0 V7/Y7777777777; x : DRAINAGE d u IM P E R V IO U S ------= 0 dx q. HOMOGENEOUS MASS AND BOUNDARY CONDITIONS b. VARIOUS MESHES c. LAYERED MASS
Fig. 1. One-dimensional consolidation of systems of porous soils ,2 J u * u - u * rf * C — ~ - 2U * U dx A2 V 3x2 l L
(u - 2ux ) * g' * cv |^ (u - 2u2 ) * S' * cv |i (11) x=o iJx=£
Finally, the governing functional in Equation 11 can be written as
A 3 = l /[a * u + * g' » cv I f - 2U « u J dx (12 ) * ry » * - 2 If] ♦ 2[(u-^)«g-cv|f] Hu.- u1) V J x=o x=£
If u satisfies identically the prescribed boundary conditions, the two
boundary terms in can be dropped. 13. If the integration over the interval (o,il) is replaced by sum of integrals over E elements, the discretized functional in Equa
tion 12 can be written as
(13)
where [N] = vector (matrix) of interpolation functions, [Nx ] = space derivation of the interpolation functions, and {qn > = nodal pore water
pressure vector. ll;. the spatial discretization is achieved by using linear and cubic pore pressure models as shown in Fig. 2. Derivation of equations for the linear pore pressure model follows. The linear pore pressure
model is (Fig. 2a)
u = [ |n - D | d * d ] {<4} (lUa)
hence, i-l. d£ dL = 1_ (N } (li+b) X dL dx a
9 o. LINEAR MODEL
NODE 1 NODE 2 •--- (L,.L2) —------—------a : ^
“= f-Ì(3"2Ll) L?L2° L2<3 -2L2> " LÌLà]^
L, = at/a , l2 = a2/ a
b. CUBIC MODEL
Fig. 2. One-dimensional element and various approximating models
10 , _ 2 , , , dL 2 where L = — (x - xQ) and — = — . Therefore e e
L=+l ± < V (V T d* = f / I fV {1,x)T dL ■ r U 1 (1 5 ) L=-l/ -1
L=+l J. / {N} {N}T = j w {N}T dL = l 0 (16) L=-l -1
Substitution of Equations 15 and l6 into Equation 13 and vanishing of variation of gives
5f*[K] {r} + [P] {r} = {R} (1 7 ) where 1 -1
[K]=I[kl, [k eJ = a e=l -1 1
E 2 1 [p] = I [Pj , [Pj = (18) e=l eJ 6 1 2 E 2 1
=1 (R > , iR } = 1 2 e=l J {<} 15. The equations for the cubic model, Fig. 2b, are
36 3a -36 e 3ae
2 ha2 -3a -a e e e (1 9 ) [ke] symm. 36 30a -3ae
Ua2 e
11 156 22ae 5 5^^ -13a
ka 13a -3ac e e (20 ) [Pel 1+20 symm. 156 -22 a
1+a^
Integration in the time domain
l6. By"writing
HH+ K4*}) ( 21 )
that is, {r^} = f2r ^ Equation 17 can he expressed as } |2rt_AtI " {rt-At} ’ t-2
(22) ([Kl + if [Pl) |rt_At [ = {E) + it [Pl {rt-4t}
17. Initial values (r(x,0)} are prescribed, hence, Equation 22 can be used to evaluate {r(x,t)} by selecting a suitable time increment At . The above integration scheme has been used since it has been often 6 7 12 13 employed for solution of similar initial value problems. 5 3 5 By making necessary rearrangements, Equation 22 is expressed in terms of increment of time factor, AT , as
c At AT = (23) a e
where At = time increment and a^ = length of an element. The time factor T is defined subsequently in Equation 25« 18. Computer codes were developed by using linear and cubic models on the basis of the following formulations. The codes were pro grammed on the time sharing system of the GE U30 computer series. The
12 banded equation set was solved by using the symmetric Gauss-Doolittle procedure.
Numerical Characteristics and Comparisons
19. The homogeneous soil system considered for this analysis is shown schematically in Fig. 1 together with the boundary conditions used herein. The closed form solution of this problem obtained by using the 9 method of separation of variables is
u: (2h)
where M = tt/2 (2i + l) , H = length of the drainage path, Fig. 1, and T is called time factor,
c t v T = (25) H
For settlement analysis from the consolidation theory, a secondary parameter called average degree of consolidation, U , in addition to the primary unknown variable, u , is defined in terms of u ,
i=o°
U* (2 6)
20. For the finite element analyses, the soil mass was dis cretized into a number of sets of finite element meshes, Fig. lb. The nodal values of u were computed by using Equation 22, and then the values of U were evaluated on the basis of Equation 26. Accuracy and stability 21. Figure 3 shows plots of percent errors in u versus number of
13 ERROR (u*-u), PERCENT lb nodes in the discretized medium. The error was evaluated as the per centage of the difference between the closed form and the numerical so lutions, u* - u , where u* is the closed form solution. As expected, the accuracy of numerical solutions improves with refinement in the mesh for both linear and cubic models. The solutions from the cubic model converge faster than the linear model in the top regions (0.1£ and 0.2£) of the consolidating mass, whereas the solutions from the linear model seem to converge faster in the middle zones (O.hü) of the layer. This behavior may be explained on the basis of the distribution of pore water pressures at various time level, T . Figure h shows distribu tions of u* for T = 0.05, 0.10, and 0.20 in comparison with the com puted values of u from the linear model with the mesh consisting of
Fig. k. Distributions of pore water pressures at various time factors
21 nodes (Fig. lb) The two solutions agree very closely. The solutions from the cubic model with 21 nodes also show similar comparisons with the closed form solutions. Generally, the rate of change of u in the middle zones is small, whereas it is relatively large in the upper zones. Hence, additional unknown nodal gradients in the cubic model do not seem necessary for the middle zone and can make the cubic model solutions slightly less accurate than the linear model solutions, Figs. 3 and 5. Figure 5 shows the convergence of the degree of consolidation
15 better agreement with the closed form solutions than the cubic model. cubic the than solutions form closed the with agreement better oain vru A fr ehwt oe eul f2, i. b It lb. Fig. 21, of equal nodes with mesh a for AT versus locations verge from below. Overall, the linear model shows as good as and often often and as good as shows model linear the Overall, below. from verge s enta gnrly byn ie atr nrmn eul o about to equal increment factor time a beyond generally, that seen is the same convergence behavior. convergence same the bv, hra te ouin fr vrg dge o osldto con consolidation of degree average for solutions the whereas above, for the linear and cubic models. The two solutions for U yield about about yield U for solutions two The models. cubic and linear the for ERROR (U *-U ), PERCENT 23. Figure 6 shows plots of errors in pore pressures at various various at pressures pore in errors of plots shows 6 Figure 23. 22. It can be seen that the pore pressure solutions converge from from converge solutions pressure pore the that seen be can It 22. i. . ovrec o vrg dge o osldto at consolidation of degree average of Convergence 5. Fig. 01 n A 0.125 = AT and 0.1 = T 16 Fig. 6. Stability of pore water pressure solutions for 21 nodes and T = 0.1
1.0, the solutions become less accurate. This is true for both linear and cubic models• The solutions for the degree of consolidation U be comes less accurate beyond AT equal to about 1.0, Fig. 7* 2k. An examination of Fig. 6 reveals that the stability behavior of pore water pressure solutions is different at different locations in the medium. For instance, in the top zones (x = 0.1£ and 0.2l) the solutions become less accurate around AT = 1.0 , whereas in the middle zones (x = 0 .hi) the solutions seem to maintain accuracy up to a value
of AT of about 3.0. 25. From the analysis of accuracy, Fig. 3, and of stability,
IT Fig. 7- Stability of average degree of consolidation for 21 nodes and T = 0.1
Fig. 6, it is seen that for complete understanding, the behavior of a numerical solution for the primary quantity u should be examined at a number of typical locations in the discretized medium. Furthermore, if a secondary quantity such as the degree of consolidation, U , is de fined for practical use, it is necessary to examine the behavior of nu merical solutions for U also. Computational time 26. Figure 8 shows plots of computational time versus number of nodes for the linear and cubic model formulations. The time shown con sists of total time for execution of a given number of time increments which includes organization of input data in the computer, evaluation of element matrices, assembly of element matrices, introduction of boundary conditions and solution of the resulting equation set. The total time required by the cubic model is generally found to be about
18 Fig. 8. Comparison of total computational times
four times that required by the linear model. 2 27. Figure 9 compares plots of the error, u* - u versus NB It can be seen that this plot gives a different view than the plot of u* - u versus number of nodes , Fig. 3. It shows that if an evaluation of accuracy versus computational effort is desired, plots such as Fig. 3 do not yield an adequate picture. 28. Table 1 shows relative errors and computational times for typical values of nodes and locations for the linear and cubic models.
19 ERROR PERCENT 800 900 Any gain whatsoever in accuracy requires significant increase in compu tational effort, indicating the tradeoffs that exist between accuracy and computational effort. Formulation effort 29. This is the effort required in formulating the problem which includes selection of mesh layout and approximating models, preparation of data, and carrying out the operations involved in the derivation of element equations. It is difficult to obtain precise time required for these operations because they would depend upon many factors including human characteristics such as skill and the experience of the analyst who formulates the problem. On the basis of limited subjective experi ence, thé formulation time for the cubic model was found to be 3 or times that for the linear model. Satisfaction of continuity of flow 30. In the foregoing analyses, consolidation of a homogeneous soil was considered. In actual field situations, however, layered soil systems are a common occurrence. In this case, the physical situation requires that the following additional conditions be satisfied at the interfaces between the layers, Fig. lc.
— + u = u (27a) m m
(27b) where = the fluid flux and is proportional to the product k^G^ » 0 = the gradient of pore water pressure, 9 m = an interface, oX Fig. lc, and the superscripts - and + denote close vicinities on either side of the interface m . The condition in Equation 27a is satisfied in the finite element formulation, since the compatibility of nodal pore pressure is guaranteed in the assembly procedure. Equation 17* The con dition in Equation 27b represents the requirement of continuity of fluid flux across an interface, and in the case of the linear model, this con dition is not fulfilled precisely. It is found, however, that although the continuity condition is significantly violated in the initial time
21 levels, it is satisfied approximately for subsequent time levels. This is illustrated in Fig. 10 for the consolidation of the three-layered system considered subsequently. The ratio = Q^/Q^ is plotted in Fig. 10 for various values of time factors, T = c -nt/H^ , where the -D V JD £> subscript B denotes bottom layer. Ideally, the ratio should be unity. It can be seen that is different from unity up to a time factor of about 3.0, whereas beyond this value of time factor, that is, for the major portion of the consolidation process, the ratio is close to unity. The values of the coefficient of permeability k^ used in computing the fluid flux Q were those obtained from laboratory exper- ±k iments, Table 2, Barden and Youhan. The approximate values of 0 were obtained as 0” - (u - u n )/am 1 , 0+ - (u - u )/am . m v m m-l" e m m+1 m" e 31. In the case of the cubic model, Fig. 2b, both nodal pore pressures and nodal gradients are compatible. Hence, in Equation 27b, since 0 = 0+ , k~ should be equal to k+ . This implies that for use of the cubic model, the system should be homogeneous. Hence, for layered systems the cubic model with nodal gradients as unknowns could perhaps provide approximate solutions and may not be suitable. 32. From the foregoing analyses of accuracy, stability, computa tional effort and satisfaction of continuity of flow conditions, the linear model can provide acceptable accuracy with most economy. Hence, for one-dimensional settlement analysis of foundations, the linear-model formulation is recommended. Two applications of the linear-model formu lation are described below.
Applications
33. The two examples considered herein are adopted from the ex- ik perimental work by Barden and Younan. These examples involve a com prehensive experimental study of one-dimensional consolidation of two- and three-layered cohesive soil systems, Fig. 11. The experiments were conducted by using a number of Rowe consolidation cells connected in series. Special care was taken to satisfy accurately the continuity conditions, Equation 27b, at the interfaces. The pore pressures at the
22 Fig. 10. Satisfaction of continuity of flow at interfaces in three-layered system (Fig. lib) interfaces were measured by using transducers, and the systems were maintained deaired by applying sufficient back pressures. 3^. Three remolded saturated clays were used in the experiments. The effects of structural viscosity were minimized by maintaining the load increment ratio Aa/a = 1 . The basic material parameters, coeffi cient of consolidation, c^ , coefficient of compressibility, a^ , and the coefficient of permeability, k^ , were determined from detailed tests for various pressure increments, Aa = 5-10, 10-20, 20-^0, and 1*0-80 psi. Only those values of c^ and k^ for the load increment 20-1*0 psi used in this study are given in Table 2 for the two- and three layered systems. The thicknesses of the component layers for the load increment 20-1+0 psi are shown in Fig. 11. 35* Figure 12 shows a comparison between numerical, experimental, and closed form solutions for pore water pressures at the impervious base 2-2 and at the interface 1-1 for the two-layered soil system, lU Fig. 11a. The closed form solutions for the layered system were ob tained on the basis of Terzaghi!s theory. Equations 1 and 2k. The re sults from the finite element analysis show excellent agreement with the closed form solution over the complete range of consolidation, and with the experimental solutions beyond a time factor value of about 0.15. The correlation between the computed, experimental and closed form values of the average degree of consolidation is considered to be remarkably good, Fig. 13. The value of At = 0.000001 year was used for the two layered system. 36. Figures 1^+ and 15 show comparisons between the three solutions for pore water pressures and degree of consolidation for the three lay ered system, Fig. lib. Again the correlation between the solutions is found to be highly satisfactory. The value of At = 0.00001 year was used for the three-layered system. 37« The computational time for solutions for layered systems in creases approximately in direct proportion with the number of elements, Fig. 16 .
2k 20 PSI 20 PSI
N O . O F N O . O F NODES NODES ELEMENTS ELEMENTS 1 r n
D E R W E N T 8 KAOLINITE 0 .8 0 ' 8 0 .8 5 " +1.25% BENTONITE v 9 1 ------1 9A D E R W E N T 4 0 .3 9 ' + 1.25% BENTONITE 13 ' iV7777777?z 8 DERWENT 0 .86" ro IMPERVIOUS VJ1 17 2 a. TWO-LAYERED SYSTEM
8 KAOLINITE 0.82*
25 3 r77/////ZA r - r 7 " " r NOTE: THE NODES WITHIN A LAYER IMPERVIOUS ARE EQUIDISTANT. b. THREE-LAYERED SYSTEMS
Fig. 11. One-dimensional consolidation tests on layered systems TIME FACTOR Tg
Fig. 12. Comparison of pore water pressure dissipation for two-layered system AVERAGE DEGREE OF CONSOLIDATION Fig. 13. Comparison of average degree of consolidation for two-layered system two-layered for consolidation ofdegree average of Comparison 13.Fig. PORE-WATER PRESSURE DISSIPATION Fig. ik. oprsno oewtrpesr dissipation pressure water pore of Comparison for three-layered system three-layered for 28
AVERAGE DEGREE OF CONSOLIDATION TIME, SECONDS 300 250 200 150 100 50 0 0
i. 1 Fig. / / © □ r ------6 10 . Computational time for layered systems layered for time Computational . -• TWO TWO -• 0 L E E R H T -0
LEGEND / / LA'S MBR F NODES OF BER UM N 20 (ER / / AYER /
30 / 30
/ • / 40 i / /
/ / / / l 50 Comments
38. The present study has demonstrated a useful finite element procedure for one-dimensional deformation and seepage in porous elastic media. A number of factors that influence the choice of a formulation scheme from viewpoints of accuracy and economy were considered. On the basis of the analyses, the following remarks are offered. 39. For the class of problems considered herein, the strategy of using lower order approximating models with reasonably fine mesh is pref erable. Specifically, the linear model is found to be better than the cubic model. 1*0. As far as engineering accuracy is concerned, the solution procedure with the proposed integration scheme would permit selection of a fairly coarse mesh. For instance, meshes with 11 and 21 nodes involve maximum errors in pore pressure solution of about 3 and 1 percent (Fig.3) and in degree of consolidation of about 20 to 10 percent (Fig. 5 ), respectively. Ul. The final selection of the mesh would, however, be guided by considerations of accuracy and economy. The pore water pressure solu tion at 0 .1 £ , 0 .2£ , and 0 .1* £ seem to become less accurate at different values of AT, Fig. 6 . The solutions for degree of consoli dation diverges beyond AT equal to about 1.0. Overall, a value of AT less than unity seems to yield a suitable mesh layout.
31 PART III: EVALUATION OF FINITE ELEMENT AND FINITE DIFFERENCE SCHEMES
1+2. Seven finite difference and finite element schemes were for mulated for the problem of one-dimensional consolidation governed by the parabolic equation (Equation l). They were evaluated quantitatively from the viewpoints of numerical convergence, accuracy, stability, and computational efforts.
Finite Difference Schemes
1+3. A number of explicit and implicit schemes were considered. The finite difference analogs according to these schemes corresponding to Equation 1 are: (i) Simple explicitiL^,1^,^,''7--l(FD)
A T ^ u ) " 0 = ut+1 - u* (28) l i i
2 "t where the symbol (6 u)^ denotes
(62u )^ = u^ - 2u^ + u* (29) i l-l l l+l
O AT = c At/a , and a = length of an element. (ii) Implicit— Crank-Nicholson^5516— 2(FD)
AT /,~2 xt+1 / «2 ,t\ t+1 t <(6 u)± + (5 u ) ^ = n ± - (30) 2
(iii) Implicit^5 — 3(FD)
Am/ j.2 ^t+1 t+1 t AT(<5 u). = u. - u. (31) 1 1 x
(iv) Special explicit— Saulev scheme. ^ ^--1+A(FD)
/ t+1 t+1 t t \ _ „t+1 t AT I u. - u. - u. + u. ) - u -u (32) V l-l l l i+V 1 1
32 (v) Special explicit— alternating direction^ ~^--^B(FD) Here the expression in Equation 32 is used at time t for mesh points starting say from left to right, and then at time t + 1 , the follow ing expression is used starting from right to left:
/ tt+2 t+2 t+1 ^ t+l\ tt+2 t+1 AT - u. - u. + u - u. (33) (ui1+1 1 1 l-i/• n J = U a . 1
kk. For consolidation in layered systems, the pore water pressures at the nodes in the vicinity of an interface were modified to satisfy the continuity of flow condition:
k 0 m m km+lÖm+l (310 where k is the coefficient of permeability, m denotes a layer, and e = f3x .
Finite Element Method
1+5- The FE formulation was covered in Part II. Two time integra tion schemes were considered. The first and perhaps the: most simple time integration scheme, l(FE), can be obtained by using the forward difference scheme {r } — ir } 1 t+At 1 tS {r} At (35)
Substitution of Equation 35 in Equation IT yields the assemblage equa tion as:
([k]+ii[P!){W }=[R]+Si[P] The second scheme, 2(FE), considered was the same as given in Equations 21 and 22. For these comparisons, only the linear approximation model (fig. l), Equation l^a was adopted. Comparisons bG. The foregoing schemes were programmed on the Time 33 Sharing system of the GE 1+00 computer. Accuracy and convergence 1+7. Figures 17a and 17b show the convergence for pore water UGCNP ------i ( r c ) — ♦ — 2 ( r o ) ------3(ro) ------4 ^ (ro ) ------4 r,(ro ) ------i (rc) ------2(FC) Fig. 17. Convergence of pore water pressures (AT = 0.125) pressure u computed from various schemes at time levels T = 0.1 and T = 0.4 for various depths in a homogeneous consolidating layer, fig. 1. Here the convergence curves are plotted in terms of percentage of error u* - u versus number of nodes, where u* is the exact solution ob- 9 tained by using separation of variables. The convergence behavior is different at different times and depths. For instance, the errors seem to increase with depth, and the accuracy improves as the time elapses. 48. Figure 18 shows convergence for the average degree of ERROR, (U* - Uî, PERCENT iuel. ovrec o ereo osldto (AT= consolidation of degree of Convergence l8. Figure 0 . 125 ) consolidation, U , at two time levels, T = 0.1 and T = 0 .k . Although the accuracy improves with time, there is no significant dif ference between the solutions from various schemes. k9. Examination of the convergence behavior of both u and U (figs. 17a, 17b, and fig. l8) shows that the accuracies offered by all schemes for an acceptable mesh, say containing 11 modes, can be con sidered satisfactory from a practical viewpoint. Stability 50. In figs. 19a and 19b are shown the plots of u* - u versus AT for time levels, T = 0.1 and 0 .h at various depths. . As it is well known, scheme l('FD) is unstable beyond the value of AT = 0.5 • The accuracy seems to improve with time for all other schemes. Scheme l(FE) and scheme 3(FD) show similar stability behavior. Although scheme l(FE) and scheme 2(FE) both show similar error (in magnitude) and stability behavior at the lower time levels, scheme 2(FE) shows im proved behavior at the higher time level. From a practical point of view, however, both finite element schemes seem to be satisfactory. The explicit finite difference schemes seem to be relatively less accurate compared with the finite element, Crank-Nicholson, and the implicit scheme. The alternating direction scheme kB yields somewhat better per formance than scheme kA. 51. Figure 20 depicts stability behavior of U at T = 0.1 and O.k . Again, scheme 3(FD) and scheme l(FE) show similar trends, whereas scheme 2(FE) seems to be less accurate than scheme l(FE). Scheme 2(FD) yields behavior similar to scheme 3(FD). Overall, the accuracy of the solutions for U seems to improve with. time. Computational time 52. The total computer times taken by each of the methods to reach the same value of the degree of consolidation, U , are compared in fig. 21. The total time includes time for input data, computation of element properties, assembly, solution of equations, and printing out of the results. The ratio of average time taken by the explicit, im plicit, and the finite element schemes is of the order of 1:U :8, with scheme 2(FD) taking about the same time as scheme 3(FD). Thus, the 36 ERROR,(U*-VÛ. PERCENT i. 9 Saiiyo oewtrpesrs (21 nodes) pressures water pore of Stability 19.Pig. ERR0R,CU^-U), PERCENT ERROR, (U* ” U), PERCENT i. 0 Saiiyo ere fcnoiain (21nodes) consolidation of degree of Stability 20.Fig. 0 5 ~ 0 4 8 0 12 10 8 © 4 2 *0 |«l ______------ I 1 1 1 I 1 1 1 1 1 . 1 1 ------______2 y^ s' — - / ------4 A (FD) A ------4 (FD) ------3 ------* + LEGEND * ----- ______* - 1 4 2( ) E (F 2 ■ 1 (FE) • ■ (FD) | F0) 0 (F 2 > 4 4 B SCHEME ------* / ( F 0) ' * * a . T = 0.1= T . a . .4 0 = T b. y AT AT 6 * — y 4 - --- ------i i l ______8 : ______ 3k ------^ L - 10 y ------ 12 TIME IN SECONDS i. 1 Cmaio fttlcmuainltms (AT0.125) = times computational total of Comparison 21.Fig. 39 any or no difference between times taken by schemes l(FE) and 2(FE). and l(FE) schemes by taken times between difference no or any schemes implicit the whereas economical, most the are schemes explicit versus the time factor T^ for the bottom layer. Results from scheme from Results layer. bottom forthe T^ factor time the versus time small a very require to found was l(FD) Scheme schemes. numerical are more economical than the finite element schemes. There ishardly There schemes. element finite the than economical aremore iue2 hw dsiaino oewtrpesrs tte nefc 2-2 interface the at pressures water of pore dissipation shows 22 Figure were Younan and Barden by conducted system soil cohesive layered interval for stable solutions and is not included herein. The layered The herein. included isnot and solutions stable for interval "- various the of application for example illustrative an as adopted system and details regarding material properties are shown in Table 2. Table in shown are properties material regarding details and system PORE WATER PRESSURE DISSIPATION The plotted. not are and 3(FD) scheme from those to similar 2(FD)were OE AE PESR DSIAIN T NEFC BTEN EWN N KAOLINITE AND DERWENT BETWEEN INTERFACE AT DISSIPATION PRESSURE WATER PORE 53. Typical results from a consolidation test on the three- the on test consolidation a from results Typical 53. Fig. 22. Comparison between numerical, closed form, closed numerical, between Comparison 22.Fig. JD and experimental results experimental and Illustrative Example Illustrative ko li+ agreement between the numerical and the closed form solution is good for all schemes. A ll numerical solutions seem to show almost the same degree of correlation with the test results, and provide satisfactory correlation with observations except during the in itial time periods. Comments 5^. The analysis presented herein is for a class of one dimensional (consolidation) problems governed by the parabolic equation. The results can be extended for other classes of problems; however, such complexities as multidimensional situations and nonlinear effects may require additional analyses. 55* The finite element schemes are based on a linear approximat ing model for u . This model satisfies continuity of u but not of . Hence, although the model can be found to be suitable for many problems, for layered media with severe changes in material parameters, some modifications in the linear-model formulation or use of a higher order model may become necessary. A formulation with modification for the continuity of flow condition in the linear-model formulation similar to the one used for finite difference procedures can be obtained. 56. Insofar as computational effort is concerned, the finite dif ference schemes seem to be superior to the finite element schemes. In the finite element schemes, computations are increased because properties of a ll elements need to be computed individually, and also larger core storage is required. From the viewpoint of accuracy, all schemes (ex cept for scheme l(FD)) can provide acceptable solutions for in itial ranges of the values of AT . 57- The finite element schemes can offer greater ease in the process of input of material properties and in handling unequal spatial mesh. Moreover, it allows increased flexibility for changing the order of approximating models. In the case of multidimensional problems, the finite element schemes can also offer ease in handling irregular bound aries . In case of nonlinearities, the finite element procedure may prove more suitable. 58. The im plicit schemes 2 (FD) and 3(FD) are relatively more accurate and stable than the explicit schemes kA and ^iB(FD). However, the explicit schemes seem to be more economical from the standpoint of computational effort. They are about k to 5 times faster than the im plicit scheme (l8). They are relatively simple to program and yield almost as accurate and as acceptable solutions as other schemes within the range of spatial and temporal discretization a user may naturally tend to adopt. In other words, although a procedure may be less e ffi cient in the higher ranges of AT from the viewpoints of mathematical accuracy and stability, within the (initial) practical range of AT , the given procedure can yield acceptable and economical solutions. For example, an acceptable solution can, be obtained from any of the schemes, except scheme l(FD), if the value of AT is adopted within unity. 59. An interesting result of this study is that for the class of problems, the simple scheme l(FE) seems to yield almost as acceptable solutions as the scheme 2(FE) used commonly in many finite element applications in the past. Qualitatively, there seems to be no signifi cant difference between the stability behavior of both the schemes. k2 PART IV: DERIVATION OF STABILITY CRITERIA FOR TWO TIME INTEGRATION SCHEMES FOR LINEAR PARABOLIC EQUATION 60. In this part, stability criteria are derived and examined for the time integration schemes indicated in equations 22 and 36. These criteria are derived by using both the linear and cubic approximat ing models, fig. 2. Linear Model 6l. We shall first consider the time integration scheme 2(FE). Following equation 22, the element equation is (37) where A = AT = c At/a2 . V Adding the element relations for two adjoining elements, fig. 23 LOCAL NODES 1 2 1 GLOBAL NODES 1 2 Fig. 23. Two adjacent elements (38a) 62. The governing equation for a generic point, say point 2, in the space-time mesh is t+1/2 (-X * i)u «1/2 ♦ (2X * |)u. l\ t+1/2 '2 3 / 3 + + 1 ■ 3 2 3 u: (38b) X*5 l6 20 Following von Neuman’s procedure, 5 5 the error can be expressed as at igx e = e e (39) ■where 3 is a real number and member of a sequence {3 } and a = a(3) n is generally complex. In order that the error shall not grow as t in creases, it is necessary and sufficient that | e0^ | <_ 1 . The expres sion in equation 38b, in terms of the error, is t+l/2 (-x ♦ i>ri/a * (- ♦ i>ri/s * (-»♦ d*3 _ 1 t ^ k t 1 t (ho) 3 E1 3 £2 3 E 3 Substituting values of e from equation 39, and rearranging leads to a(At/2) 2 X ♦ | + ( - X * i)(e -16a ♦ e1Ba) = | ♦ 1 (e-iBa + e16*) (1.1 ) Therefore, the stability condition is - At _ h + 2 cos 3a (U2) h + 6X + (2 - 6A) cos where 0 Q -i3a . if 2 cos 3a = e + e Cubic Model 63. The assemblage equations for two elements are f ~ 36 3a -36 3a 0 0 ~156 22a 5^ -13a 0 0 2 3a ha.2 - 3 a -a 0 0 22a Ua2 13a -3a2 0 0 -36 -3a 7 2 0 -36 3a + - Ì - 5b 13a 312 0 5b -13a < —30 2 2 210 3a -a 0 8a2 -39 a -13a -3a2 0 8a2 13a -3a2 0 0 -36 -3a 36 -3a 0 0 5b 13a 156 -22a 2 0 0 3a -a -3a Ua2_ _ 0 0 -13a -3a2 -22a > a 2J t+1/2 ^156 22a 5^ -13a 0 0 22a Ua2 13a -3a2 0 0 1 5h 13a 312 0 5b -13a ~ 210 -13a -3a2 0 8a2 13a -3a2 0 0 5b 13a 156 -22a - _ 0 0 -13a -3a2 -22a k & 2 ~ ^ S3. In this case, there are two equations at point 2 that need to be examined. 3a0^ + 72u 2 - 36u ^ + 3a0 j 30 ( * 3S - / -t+1/2 + 210 ( 5lmi + 13a6i + 312u2 + 5ltu3 _ 13a03) t + 1 j2 1 210 (5^ + lSae^^ + 312^ + 5^u 3 - 13a93j (UUa) and A__ 2 2 2 \ a 0 + 8a 0 - 3au^ - a 0 J 30 (3aui - ± C. 3 3 /, t+l/2 2 ^ 0 " (-lSa^ - 3a291 + 8a202 + 13au3 - 3a20. ) t+1/2 1 '^-1381^ - 3 a 2 d1 + 8a202 + 13au3 - 3a203) (UUb) 210 By using the expression for error, equation 39> 0 = |$- ieeatei6x d X (^5) By following the von Neumann’s procedure, the criteria corresponding to equation UUa is a(At/2) N (U6) e D where / -i6a i$aNj 13a . Q|( -i6a i6a\ Ve + 6 ,' + 210 [e ~ e ) A 36x 5U / -i6a i6a -(e_i6a + dUa) + (e + e / ~ 30 V / 210V -i6a . i8a\ •O !3a/ --i6a _ i6a\ » + + e ) ’ie 6 ) - 210\ -i6a ~i6a , -i6a Substituting e"1"^0, -- e = 2i sin 6 6aa and e M + e M = - 2 cos 6a 2 and i = -1 and rearranging • a(At/2) ______312 + 108 cos 3a + 26a8 sin-ga ------< x /i>7_\ e “ (312 + 5 0 UX) + (108 - 50UX) cos 6a + (26a8 - k2aX$) sin 8a - V'+l-a; Similarly for equation ^Ub: 8a8 + 26 sin 8a - 6a8 cos 8a < 1 (v n > ) 8a8 + 56a8X + (26 - U2X) sin 8a - (6a8 + lHaftX) cos 8a 6h. The time integration scheme used above is based on the value of u at midpoint of At as an average of values at t and t + At , equation 21. This scheme is used often in FE applica tion.^ ’ ^ ’ 12,13 As seen in Part III, the simple scheme, l(FE), equa tions 35 and 36, that is based on Euler type integration in time is also suitable for successful applications. In the following are derived sta bility criteria for the scheme 1(FE). The element equation with linear model, according to this scheme is By following the foregoing procedure, the stability condition for a scheme l(FE) is | * 1 (e« a . e-i8a) aAt e (^9a) | + 2i - x(el6a + e_iSa) + i (e16a + e"16“) Therefore, ______k + 2 cos 3a (U9b) 1* + 12X + (2 - 12X) cos ga - 1 Similarly, the two criteria corresponding to u and 0 , respectively, in the cubic model are ______312 + 108 cos ga + 26a sin ga______(50a) 312 + 1008X + (108 --1008X) cos ga + (26ag - 84aXg) sin ga - 1 and ______8ag + 26 sin ga - 6ag cos ga______(50b) (8ag + 112agx) + (26 - 8hX) sin ga - (6ag + 28agA) sin ga ~ Discussion 65. The stability criteria in equations h2 and ^7 for the scheme 2(FE) and in equations *+9 and 50 for the scheme l(FE) are satisfied for all values of X >_ 0 . Hence, mathematically the schemes are uncondi tionally stable. As implied in Part III, however, notwithstanding un conditional stability, the suitability of a scheme will be governed by the accuracy it can provide within the ranges of discretization the user tends to adopt. PART V: FINITE ELEMENTS IN THE TIME DOMAIN'21,22,23 66. A time-dependent problem with the FE method is usually- solved in two stages. First the problem is discretized in the physical, space and, in the next stage, solution in the time domain is obtained by using .an integration scheme. The latter is based on FD procedures. 8 The procedure can be expressed as u(x,y,z,t) = [N (x ,y ,z) ]{q(t)} (51a) S and iq(t)} = [Nt (t 3At )]{q(t-At)} (5113) where u is the unknown, N ^ are interpolation functions in terms of x, y, and z, (q(t)} is the vector of nodal unknowns as function of time, and are interpolation functions in the FD schemes for the time domain. 67. Instead of expressing u in two stages as above in the pro cess of complete discretization, u can be expressed in terms of inter polation functions Ng^ that include both space and time (u(x,y ,z,t)} = [Ngt (x,y,z,t)]{q} (52) This concept has been proposed by Oden,2*1 Nickell and Sackman,2^ 26 27 Fried, and Argyris and Scharpf. It has not been fully investigated; some preliminary comments and works, Zienkiewicz and Parekh,"^ 28 Q po PH Zienkiewicz, Desai and Abel, McCorquodale, Bruch, have been reported. 68. One of the aims of this project was to investigate the con cept of complete discretization and examine.its relative advantages or disadvantages in relation to the procedure, Equation 51. 69. Although the concept is elegant and may prove beneficial for certain problems, it seems that it may not be attractive from viewpoints of computational and formulation efforts as compared with the 1+8 semidiscretization procedure. This belief will, however, await results of further investigations. Unfortunately, since this project was post poned, only preliminary formulation work can be performed. 70. The work accomplished is as follows: a. Formulation of space-time concept for dynamical systems with a quantitative example for single-degree-of-freedom system. b_. Formulations of elements in time for fluid flow problems: (1) One-dimensional fluid flow or diffusion governed by a nonlinear equation. (2) Two-dimensional fluid flow governed by a linear equation. Dynamical System^ 71. The description below follows the works in references 24, 26, and 27. The variational principle used commonly in dynamics is Hamilton!s principle. The functional in this principle is the Lagrangian L L = T - (U-V) +W (53) where T is the kinetic energy, U is the strain energy, V is the work done by external forces, and W is the work of viscous forces. Variation of L in Equation 53 yields 6L (T - U + V + W)dt = 0 (54) where t^ - % denotes the time interval of interest. 72. Let u denote the unknown displacement and be expressed through interpolation functions, N , Equation 52. The kinetic and strain energies, Tg and Ue , for a space-time element can be ex pressed respectively as Te = |- iû}T [m]{ù} (55) 49 where [m] is the element mass matrix and the overdot denotes deriva tive with respect to time; that is, (ù) is the vector of instantaneous velocity, and Ue=| {u}T [k]{u} - {u}T [w]{p} (56) where {u} is vector of displacement, [k] is the element stiffness matrix, and {p} is the vector of external forces, and [w] is the work matrix. Work of the viscous forces can be expressed as W = {u}T [c]{u} (57) e where [c] is the element damping matrix. 73. By using Hermitian interpolation in the time domain u(x3y,z,t) = u1N1 + u^N^ + + ^2?*k = [N]T {q} (58a) and u(x,y5z3t) = u1N1 + u 2N2 + + u^N^ = [N]T{q} (58b) where {q} is the vector of nodal displacements and velocities, and 3 2 N = 2T° - 3T + N2 = -2TJ3 + 3T 2 (59) N 3 = (T3 - 2T2 + T) At = (T3 - T2 )At where T = t/At , At is length of element in time and t is the time level. Jk. Substitution of Equation 58 in Equation 53 yields 50 L = \ J ({q}T[N][m][N]T{q} - { + 2{qJT [ N ][c ][n]T{q_} + 2{q}T [N] [w][Kf ]T {pn ) )dt where is given as ip) = [Nf](pn} (61) and {p^} is the nodal force vector. 75 Equation 60 over time At leads to = w)t ( [k][A2 ] + [c ][A3 ]){q> + {q.}T [w][Alt]{pn } (6 2 ) where * 36 -36 3 3 -36 -3 -3 1 (6 3 a) A i = 30At Symm. 1+ -1 k - 156 5U 22 -13 156 13 -22 At A 2 = ÏÏ2Ô (63b) U -3 Symm. k 51 30 30 6 -6 -30 30 -6 6 A - At 3 " 6o -6 6 0 -1 (63c) 6 -6 1 0_ ’ 21 9" 9 21 3 2 -2 -3 in which ^Nf]T = [l - T T] 76. The element equations, Equation 62, can he assembled to ob- tain the global relation as L = \ {r}T [M]{r} - | {r}T [K]{r} + {r}T[c]{r} + {r}T [W]{P} (6M where {r} is the global nodal displacement vector, [M] is the global mass matrix, [K] is the global stiffness matrix, [C] is the global damping matrix, [W] is the global load matrix, and [P] is the global nodal force vector. By taking the variation of L in Equation , the assemblage equations are obtained as [M]{r} - [K]{r} + [c]{r} + [W]{p} = 0 (65) Application 77- Consider rather an elementary example of the single-degree- of-freedom system, Figure 2^-a. A computer code was prepared on the basis of the foregoing formulation for the spring mass system. The equations governing the displacement u of mass m under a transient forcing function f(t) is mu + ku = f(t) (66) 52 H 35 a. Spring mass system f(t) - 1 b. Forcing function Fig. 2k. Single-degree-of freedom system where k = stiffness of the spring. The closed form solution to this equation for f(t), shown in Figure 2klo, is u = u (1 - cos pt) st (67) where p = k/m . For example, u = 1 , f(t) as in Figure 2Vb and 2 st p = 1 were adopted. 78. For the space-time FE analyses, a distance in time equal to 2tt was considered and was divided into meshes with number of ele ments NEL = 20, 10, and k. The results are shown in Tables 3-5* It can be seen that the FE results are very close to the analytical solution. 53 Fluid Flow Problems One-dimensional flow: nonlinear equation 79. The governing nonlinear (Bossinesq) equation is expressed as19, 30 (6 8 ) where k = coefficient of permeability, h = height of water table, n = porosity, x = horizontal coordinate, and t = time, consider the problem of flow in and out of a rectangular (earth) bank, Figure 25. t =0 Fig. 25 The boundary conditions associated with this problem are h(x,o) = hQ = h2 h(o.,t) = h(“ ,t) = h2 The bottom and the right hand side boundary can be considered to be impervious. Formulation 80. The Galerkin residual procedure was used to derive the FE equations. This procedure is found to he more general and unlike the variational approach, does not need existence of a variational func tional. In the Galerkin method, the residual for the governing equation is formed. The interpolation functions used to approximate the unknown variable of the problem are used as weighting functions in the Galerkin method, and the weighted residual R is equated to zero. Thus, R is (69) Using an isoparametric quadrilateral space-time element, fig. 26, h is expressed as (TO) where = 1/4(1 + ss^)(l + tt^) , s , t = total coordinates of the element, Figure 26. The functions are such that their values are h L Fig. 26 unity at node i but.zero elsewhere at all other nodes. 8l. According to the Galerkin method dxdt = 0 (71) or 55 (EYJ)(EVJ)}axat ♦ J / M ’V j - //*i It K hj)to dt - ° Integrating the second term by parts and cancelling appropriate terms, __3N. fS 3x ) h (eY j) at a dx dt //* i at (73) N. l k (“ /j) * - ° 82. Equation 73 represents a set of nonlinear equations that can be solved by using Newton-Raphson type procedures. 83. Galerkin formulations based on a linearlized version of Equa- tion 68 have been obtained (Desai ) by using the semidiscretization approach. It will be useful to compare and evaluate the foregoing (complete discretization) procedure with the results from the previous method. Moreover, it will be worthwhile to evaluate the procedure from viewpoints of numerical convergence and ability. Two-dimensional flow 8U. Finite element solutions for two-dimensional unconfined seepage based on the linear equation + v 32Q k ¿a (7U) and the semidiscretization procedure have been obtained previously.^ Here 0 = fluid potential and Q = applied fluid flux. The problem can be formulated by using a space-time element as described in the fol lowing . Let 56 0 = ENi0i = [Nj {g} (75) Then, according to the Galerkin method (76) D - ) - Q Si dx dy at = 0 J = 1, 2, ...M Where M *= number o f nodes in the d isc r e tiz e d m ass, and D = flow domain. Use of Green’s theorem leads to 9N. 9EL 9N. 9N_.9N. 3N.\ ///ï^x 9x 9x + 9y 3T “ j « 7 f f j - J f a f f a ti + ky tj) ndS = 0 (77) D S i “ 1 which in matrix notation can be written as [k]{q} = {Q} (78) where dN. 3N. dB . 9N. 9N. \1 [k] = / / /lfkx 3^-+ ky ÿ (79a) / / t L 9T + **i ^ / J D and M {q) - at +ffa | [(kx ix ♦ y ^ - *y)] nds„< (798) D S i = l Here S - part of the surface, and ¡L , £ , SL denote unit normals x ’ y ’ z to the surface. 85. Interpolation functions, EL , of different orders can he 57 employed to examine economy and mathematical characteristics of the space-time approach. These can then be compared with the corresponding properties of the semidiscretization procedure. 58 SECTION II fcy J . T . Oden PART VI: SOME ASPECTS OF THE THEORY OF FINITE ELEMENT APPROXIMATIONS OF THE DIFFUSION EQUATION* Introduction 86. This portion of the report is an exposition aimed at analyzing a number of techniques for arriving at error estimates of finite-element/ finite-difference approximations of certain time-dependent problems. We shall emphasize linear diffusion equations for two reasons: first, to keep the scope of the discussion within reasonable limits; second, because the underlying theory is more completely developed for this class of problems. Much of what we consider can be applied to finite- element approximations of all types of linear boundary-/initial-value problems. 87. In the exposition to follow, we describe a number of basic techniques for determining rates-of-convergence of approximations of a class of linear time-dependent problems. We choose to categorize these techniques as follows: a. Semigroup theoretic estimates. It is a widely known re- ~~ suit in the theory of partial-differential equations that for a broad class of evolution equations, the fundamental solution operator is a member of a semigroup of operators.3^,35 By using certain basic properties of semi-groups, error estimates can often be easily obtained, particularly in those cases in which it is possible to transform the problem into a one-parameter family of elliptic problems. This makes the problem of estimating the spatial rate-of-convergence relatively straight forward. The rate-of-convergence of the temporal approximations can be established by exploiting certain properties of fundamental solutions. For example, if the fundamental solution is a member of a semigroup, then it has a matrix exponential form which can be approximated using the Pade^ matrix-approximation theory. The use of semigroups in difference approximations of time-dependent problems has been discussed by several authors, Peetre and Thomee,^ and Widlund.37 Some features of the methods * This portion is based on a part of the paper by the author and L . C . Wellford. 33 6l we discuss were used by Babuska and Aziz in collaboration with Fix.38 b. Energy methods . In most mathematical models of physical problems a norm can be developed which is equivalent to the total energy in the system. Normally, the study of convergence of various Galerkin approximations in an appropriate energy norm is a very natural undertaking. This is due to the fact that the weak forms of most boundary-value problems of mathematical physics can be interpreted physically in terms of changes in energy. The use of energy methods can be found in the finite- difference literature39 and variants have been used by Fujii^O and Oden and Fost1^ for finite-element analyses. c_. L2 methods. A fairly extensive literature has aecumu- iated in recent years on Galerkin approximations of the diffusion equation in which L2~estimates in spatial variations and L2- or Loo-estimates are obtained in the temporal variations. This "L2-theorv" has been largely developed by Douglas and Dupont,Varga,^ Wheeler, ^5 and others, Interestingly enough, the methods do not involve energy-error estimates and enable one to go directly to stronger L2~estimates. 88. There are, of course, several other techniques in use for studying finite element approximations of time-dependent problems. The 1*6 projection methods of Thomee, for example, make use of the fact that the finite-element technique produced a system of difference equations in Rn . By using standard difference concepts and projections from Rn back into the space V in which the original problem is posed, error-estimates for certain finite-element approximations can be obtained. Alternately, certain parabolic problems can be shown to have coercive properties under an appropriate choice of norm (see, e.g., Lions and Magenes ). Thus, "elliptic-type" error estimates can be obtained, h8 as shown by Celia and Cecchi. It is only a matter of interpretation as to whether or not these "projection" and "coercive-operator" tech niques do not actually belong to the semigroup and the energy methods described previously. 89. As in elliptic theory, the study of convergence of finite element approximations rests firmly on certain results from interpolation theory. For this reason we discuss in the section following this intro duction certain features of interpolation theory which are essential for 62 our investigation. Next, in Section 3, ve describe finite-element/ Galerkin models of a general class of diffusion problems, and in Sec tions U, 5, and 6 we obtain error estimates for these models using the semigroup theory, energy methods, and Lg-methods, respectively. Finite Element Approximation and Interpolation 90. Consider a Hilbert space H whose elements are functions u(^) of points £ = (x1 , x2 xn) in some bounded domain ft of Rn . In subsequent sections, the context shall make clear the specific properties of H , but for the moment we need only assume that it is endowed with an inner product, (u^,u2) . A finite-element model of ft (and H) is another region ft which is partitioned into a finite number E of disjoint open sets ft^ called finite elements: E ft = I I ft ; ft ft _ = 0 if e ^ f (8 0 ) \J e e i e=l Here ft is the closure of ft . Within each element we identify a set e a, , of local basis functions t|t~'e'(x) which have the property that g a ga M e a(e) (e) D t N ( 4 - s" V f % (?} = 0 <3,g e Z* ; M,N = 1,2,...,Ne ; e,f = 1,2,. ..,E ; |a| <_ k (8l) got M Here x^ is a nodal point labelled M in element flf • S • SK • 6e , are Kronecker deltas, N is the number of nodes in element ft , f 5 e e and we have used multi-index notation; i.e., a and g are ordered n-tuples of non-negative integers, a = »•••* an ) * and following conventions are used: 63 _ot / \ _ 3'“ 'u(x) -- : a = an + a_ + . .. + a D-u(x) = ------— a 1 ~1 1 2 n . a_ an n 3x_ 19x02. . . 3x 1 2 n ol a_ a 0 a ß aß a ß a _ 1 2 n .aß x- = x x . . .x ; ö~~ 6 1 X6 2 2 . . .6 n n (82) 91. The local representation of a function in terms of the basis a (0 j functions (x ) is of the form N ~u / \ N ^ e V (x) = £ ^lr6 cla » / _ \ \jj~ ,“(S> (?) » a (?) = D (83) e Ia I E G V(x) = E V (x ) = E E a \“(x ) (810 e=l 6 I a I Here x~(x) are global basis functions given by A Ev N e (e) / x a, « (x) = E E ft Y ie,(xJ (85) e=l N=1 A N ~e (e) where 9 ^ defines a Boolean transformation of the disconnected system of elements into the connected model £1 |i.e., 2^=1 if node N of coincides with node x^ of 9 and 2^ = 0 if otherwise) . e ~ ( a, ^ G 85. Suppose 9 = 9 . Then the set of functions 5 |a| 6h global representation is of the form N V(x) = £ AN 92. Returning now to the space H , consider a typical element u = u(x)) . The pair {(*,*) define an orthogonal projection : H-+S^(Q) such that N Q,u = W = £ ( ta, E(x) = u(x) - Q^u(x) (88) is referred to as the (pointwise) interpolation error of the finite element approximation W(x) = Q^Cx) • Its properties depend explicity on the properties of the subspace . 93. To appreciate the importance of the interpolation error in finite-element/Galerkin theory, consider the abstract boundary-value problem, find uetf such that (P(u),v) - (f,v) v e M (89) where P:H->M is a linear operator. The finite-element/Galerkin approxi mation of the solution u , Equation 89, is the function UeS^ such that (P(u),v) = f,V) V e Sk(a) (90) The function 65 e(x) = u(x) - U(x) (91) is the (pointwide) approximation error of U(x) . Since U(x) e S fc(fl) there is a mapping II H+Sk(a) such that n^u = U ; but is not __ „ ^ n ' f n / ' / \n\ an orthogonal projection into relative to j* Indeed’, hy setting v = V (Equation 89) and subtracting (Equation 90) from the result, we see that (Pe,V) = 0 ; thus, Pe is orthogonal to h * 9^i. The function e(x) = n^u(x) - fyu(x) = U(x) - Qju(x) (92) is referred to as the projection error. In most of the developments to follow, we show that it is possible to bound certain norms of the pro jection error by the corresponding norm of the interpolation error; i.e., we derive relationships of the type ||e||tí = C(h)I|e ||h (93) Since e = E-e , use of the triangle inequality gives | |e | |H < (1 + C(fi))| |E| |h (9b) Thus, (i) if the coefficient (l + C(k)) remains bounded as hr>0 , and (ii) if | |e | |^-K) as k+Q , we have proved convergence of the method in the | |* | norm. Criteria . (i) is a question of stability of the approximation, while (ii) is a question of consistency. The latter question depends explicitly on H and the properties of S^(Q) , so that the convergence of the method is connected to the interpolation error in a fundamental way. 95. In many instances, the space H is a Sobolev space H 331^ ) , the elements of which are functions' whose partial derivatives of order <_ m are square integrable on Q . The inner-product in Hm(ft) is then 66 (u,v)m J Z D~uD~vdft (95) 2 | a | < m and the norm is (96) where di2 = dx.dx„...dx . In such cases we construct the finite- 1 2 n element subspaces S^(ft) so as to have the following properties: (i) For every ueH111^) , there is a constant C such that I|Qfcu||m < C||h||m for all k>0. (ii) If p(x) is a polynomial of degree <_ k , Qhp(x) = p(x) (iii) Let hr>0 uniformly (i.e., for each refinement of the mesh, let the radius of the largest sphere that can he inscribed in he proportional to k^) . Then there is a constant K independent of h such that k+l-n u |i e |I n < Kfc k+1 (97) for n < m , where lul. is the semi-norm — 11k+1 2 u (98) m li9 Interpolation results such as (Equation 97) were derived hy Strang, Ciarlet and Raviart,^ and others. We shall henceforth assume that the spatial interpolation spaces S^(ft) have properties (i)-(iii). 67 Finite Element Approximation of the Diffusion Equation 9 6 . We now consider a class of time-dependent problems charac terized by equations of the form ” "f*-— + A(x)u(x,t) = f(x,t) X £ Q D~u(x,t) = 0 , % e ; |a| <_m - 1 u(x,0 ) = u q (x ) , x e ß (99) ■where A is "the 2m--th order differential operator A(x) = Ï (-1) .(x )D^ (100) ' |a |, | g | < m 26- ' and the coefficients A (x) are such that A is m-elliptic (strongly % 9 « elliptic); i.e., there exists a sesquilinear form a(u,v) = (Au,v) = j £ A (x)D~uD~vdft (lOl) i,21 ot j , | g | such that there are positive constants y and v1 for which a(u,u) >_ uQ (102) and a(u,v) £ y]_| IuI |m | IvI |m (103) 68 We then replace Equation 99 by the equivalent (weaker) variational problem, (^VLd[t- >v)0 + a(u(t),v) = (f(t),v)0 (u( • ,0) ,v)Q = (u q ,v )q , v e Hjj(fl) t e (0,t] (10U) where is the Sobolev space of Hm (C2) functions with-compact support in . 97. Now consider a Galerkin approximation of Equation 103 which involves seeking a function U(x,t) e S^(ii) x C (0,T) such that ^ M t l ^ + a(u(t)>V) = (f(t),V)0 t e (0,t] (U(-,O),V)0 = (u0 ,V)Q T V e S h(Q) (105) Since a continuous dependence on t is still assumed, U(x,t) is referred to as a semidiscrete Galerkin approximation. Now, the coef ficients AN are functions of time t . Thus, Equation 105 leads to a system of first-order differential equations for the specific coef ficients AW (t) corresponding to the finite-element approximation + ¿JW (t> - V t( M=1 N„ = l. (106) 'N M=1£ ,w M<°> - Here 'NM ’ f'lIM f H = «V m ^6tUn ,V^0 + 0a(Un+1,V) + (l-0)a(Un ,V) = (f(t),V)Q t e (0,t] (u°’v)o - (v v)o * T e (107) where 6. denotes the forward difference operator; i.e., <5 l/1 = , t ^ (Un 1 - Un )/At Evaluation of Equation 106 with the finite element approximation, Equation 86, leads to a system of algebraic equations for the coefficients = A^(tn) : " M?°t> + + 4tA A (108) Error Estimates for the Diffusion Equation Using Semigroup Theoretic Results 99. The calculation of error estimates for equation 10k can he embedded in the theory of the approximation of elliptic partial differ ential equations by using semigroup tools. In particular, the 70 traditional characterization of the semigroup through the resolvent operator can be used to transform the parabolic diffusion problem into a one-parameter family of elliptic problems. The error in the spatial discretization for the parabolic problem can then be related to errors encountered in modeling elliptic problems. On the other hand, errors due to’the discretization in time can be determined through the expo nential function representation for the semigroup and its relationship to the Pade'' approximations. 100. Initially, we must define the components of the error of the approximation scheme. u(x,t) is the exact solution to Equation lOU; U(x,t) is the solution to the semidiscrete Galerkin approximation (105); and t/^x) is the solution to the finite difference-Galerkin approxima tion, Equation 107* at time t = NAt . We introduce the definitions, e(x,nAt) = u(x,nAt) - Un (x) = approximation error (109) a(x,nAt) = u(x,nAt) - U(x,nAt) = semidiscrete approximation error (110) x(x,nAt) = U(x,nAt) - Un (x) = temporal approximation error (ill) Then, for any choice of norm on x , Ie(t ) I I = I |u(•,nAt) - U n (*) +U(-,nAt) -U(*,nAt)|| (112) 101. The evolution problem, Equation 99* is reminiscent of the linear dynamical system § - Aq. = f , g(0) = qQ the solution of which is 71 For simplicity, let us assume that f(s) = 0 . Then the matrix E(t) E(t) = e~ is called the fundamental solution operator and q(t) = E i t ) ^ The operator E(t) is a member of a multiplicative semigroup, G ; i.e., if E1 (t) and E 2(t) are in G , then the semigroup properties, (i) E^ *E2 e G (closure) (113) (ii) E1 *(E2 «E^) = (5’L,?2^*?3 (associativity) are satisfied. In fact, we also have the important properties, E(t)E(s) = E(t + s) lim t-HD E(t) = I (11^) 102. How does one construct the fundamental solution operator E(t) from a given matrix A ? Let L(q(t)) = g denote the Laplace transform of g(t) . Then sq. - q(0) = Ag Thus, if s is not an eigenvalue of A , g = (sI-A)-1^ E(t) = L-1(sI-A)-1 (115) 103. Now a similar situation is encountered in the use of the 72 weak formulation of partial differential equations of the type as in Equation 10k. In this case we define a fundamental solution operator E(x,t) such that u(x,t) = E(x,t)uQ (x) Then it is clear from Equation 115 and the definition of the resolvent operator R(x,s) = (si-A) ^ that R(x, s S^E(x,t)dt (116) Hence, if u = L [ u ] u(x,s) = R( x ,s )Uq (x ) and, of course, u(x,t) StdsuQ (x) (118) where V is a contour in the complex plane. Now we take the Laplace transform of the weak parabolic partial differential Equation (10k) and set f(t) = 0 , for convenience. Then ^3u(t) (119) ^ 3t , V)o J + jL[a(u (t ) ^ ) ] = 0 V v e H^(ft) Now L = s )v(x)dft / u.Q(x)v(x)dft = s(u(s),v)Q - (uQ ,v)0 ( 120) 73 and i-[a(u (t),v)J = a(u(s),v) (121) Hence introducing Equations 120 and 121 into Equation 1199 we obtain the boundary-value problem s(u(s),v)0 + a(u(s),v) = (uQ ,v)0 v e (122) 10^. Similarly, if U is the Galerkin approximation to the exact solution, then we define the approximate fundamental solution operator E, (x,t) so that ~h ~ U(x,t) = E^(x,t)Uq (x ) (123) Then we introduce the approximation A^ to matrix A . And an approxi mate resolvent operator R^(x,s) is defined such that Rh (x,s) = (sI-A^)-1 Then the approximate resolvent is related to the approximate fundamental solution operator through - rl[(si-Ah)_1] - rl[?h Thus, 00 R (x , s) = / e-S^E (x,t)dt (l2U) ~ h o ~n Hence if U = l[u] , then the transformed version of Equation 123 gives U(x,s) = R^(x ,s )u q (x ) and, of course, 7k U(x,t) dsuQ(x) (125) 105. Now if we take the Laplace transform of the semidiscrete Galerkin equation (105) and set f(t) = 0 for convenience, we obtain an approximate boundary-value problem for (122) s(U(s),V)0 + a(U(s),V) = (u0 ,V)0 V VeS^fl) (126) Now subtracting (126) from (122) and defining the transformed approxi mation error by u(s) - U(s) = a(s) = l[a(t) ] = L[u(t) - U(t)] s(a(s),V)Q + a(a(s),V) = 0 ¥ V e Sfe(fl) (127) Thus it is clear that if we can estimate the approximation error a(s) for the boundary-value problem, Equation 1 2 2 , then we can deduce the approximation error for the true problem, Equation 10U, using the in verse Laplace transform, Equations 118 and 1 2 5 - cr(t) = i-”1 [ct(s )] = L"1 [u(s) - U(s)] = / ig(s?s) - Bh (x,s)] estdsu (g)' (128) i r To insure that the inverse Laplace transform exists, we assume that the operator A is the infinitesimal generator of a strongly continuous semigroup. The Hille-Yosida-Phillips theorem (Friedman^ ), guarantees that if the operator A is the infinitesimal generator of a strongly con tinuous semigroup, there exist real numbers M and oj such that for every R^s > m , s is in the resolvent set of A (i.e., (s I-A)-"^ exists) and ||R(x,s)n || <; --- n = 1,2,3,... (129) (s - (b) Thus if we select the contour T in the complex plane so that if s e r , R^s = positive constant and R^s > a) , then F lies in the right half 75 plane and the integrand in Equation 128 is bounded in accordance with Equation 129 as s increases. Thus the Laplace transform^ Equation 1 2 8 5 exists. 106. The operator A satisfies the m-elliptic condition, Equa tion 102. Thus the operator -A + R^sl occurring in Equation 122 also satisfied an m-elliptic condition since ((-A + Resl)v,v)0 = -(Av,v)0 + Res(v,v)0 = a(v,v) + Rgs(v,v)Q = a(v,v) + Rgs| |v| |q (130) ^ a(v,v) Thus the boundary values problem, Equation 122, is strongly elliptic for each s e T . Equation 130 lead to the following error estimate. Theorem k.l. If U(s) is the Galerkin approximation in to u(s) , the solution of Equation 122, then R s ) - U(s)||m * C1hk+1"m|u(s)|k+1 (131) Proof: From the m-elliptic property of operator -A + R^sl » Equation 130, we have that if U*(s) is an arbitrary element of H0l|u(s) - U(s)||^ £ Res(u(s) - U(s),u(s) - U(s)) + a(u(s) - U(s),u(s) - U(s)) = Rgs(u(s) - U(s),u(s) - U*(s) + U*(s) - U(s)) + a(u(s) - U(s), (132) u(s) - U*(s) + U*(s) - U(s)) = Res(u(s) - U(s),u(s) - U*(s)) + a(u(s) - U(s),u(s) - U*(s)) To obtain the last result, we have used the identity 76 Res(u(s) - U(s)?U*(s) - U(s)) + a(u(s) - U(s),U*(s) - U(s)) = 0 obtained by taking the real part of Equation 127 and setting V = U*(s) - U(s) . Now using the Schwarz inequality, Equation 103, in Equation 132 U0||u(s) - U(s)|£ £ EgSuJuis) - U(s)||0||U*(s) - U(s)||o + ijJR s ) - U(s)||J|u*(s) - U(s )||iA (1 3 3 ) * (Res ^ 2 + PQ)||u(s) - U(s)||m |ju*(s) - U(s)||m Thus simplifying Equation 133 R sp.p0+n- IR s ) " U(s)||m £ — ||u*(s ) - U *(s )||m (1 3 ^ ) Now let U*(s) be the arbitrary element of which interpolates u(s) . Then we see from the interpolation result, Equation 97, that R sp- |i 0+ lu , -i |u (s) - u(s)||m £ - — ------kh 1_m||n(s)||k+1 k+l-mii = C^ti 'k+l Thus the error estimate for the semidiscrete approximation, Equation 105 to Equation 10^-, can be determined. Theorem 2. If u(x,nAt) is the solution to Equation 10k a t tim e t = nAt and U(x*nAt) is the solution to Equation 105 at time t = nAt , then 77 Proof: Using the transform (Equation 128) between the error involved in the approximation of Equations 10^ and 122 the result follows ||o(x,nAt)||m = ||u(x,nAt) - U(x,nAt)||m = M u(s) “ u(s)) eStdsllm - C .k +l-m r M—/ mi s t , S 2^7h / rllu(s )'fk+i e ds - , k+1 -mil / , mi = C2h ||u(x,t)||k+1 107. Now le t E^(x;t,T) be the fundamental solution operator associated with the semidiscrete Galerkin approximation (Equation 105) At 0 and E^ 5 (x;(u+l)At,uAt) be the fundamental solution operator (often called the amplification matrix) associated with the finite difference-Galerkin approximation (Equation 107). Then U(g,(u + l)At) = E^xiOj + l)At,uAt)U(x,uAt) (136) and + 1 (x) = EhAt,9 (x;(u + l)At,uAt)Un(x) (137) At 0 The operators E^ and E^ 9 have the semigroup property of Equation llH . Thus E^ixjtjT) = Eh(x;t,s)Eh(x;s,T) (138) EhAt,0(x;uAt,1lAt) = EhAt’ 9(x;uAt,7At)EhAt,e(x;7At,TlAt) 7] 7 £ u (139) 78 Now the semidiscrete problem in Equation 105 is assumed to be well- posed and the finite difference-Galerkin problem in Equation 107 is as- summed to be stable (depends continuously on the initial data). Thus the fundamental solution operator and the amplification matrix are bounded l|Eh (x;(u - l)6t,0)U(x)||m ^ Cj|u(x)||m u = 0,1,.. .N m 3 ~ m ¥ U e sh (fl) (l40) E^t,0(x;nAt,vAt)V(x) ||m £ qllvix)^ 0 <_ v <_ n <_ i ¥ V e S, (0) (l4l) h Equation (l4l) implies that E At,6 ||EhAt,6 h ’ .(S;nAt.uAt)v(g)|| (x;nAt)|| sup m VeSh (fi) lv(x)f|m ^ q 0 <; u £ n N (1^2) 108. Consider the semidiscrete finite element-Galerkin Equation 106. Solving Equation 106 for the value of the solution vector AM (uAt) based on the "initial value" of A^(u - l)At) and setting f = 0 AM (uAt) = E ]y[pA P ((u - l) At ) (143) where E^P = and -1 BM _ ^MTpSjp__7] ' The matrix EMp is a dis- crete approximation for the fundamental solution operator E Sim- 'h*II ilarly solving the finite difference-Galerkin Equation 108 for the value of A^ in terms of A^"1 and setting f_ = 0 n 9 A U_1 \ = F ' - A (144) where Mp [6M7 + M < 1 - 6)6m » V 1[6 7p 79 The operator F..Q in Equation lUU is a discrete analogue for E. At,0 . Mp h 109» A useful approximation for the term in Equation ll+3 can be obtained through the Pade approzimations. The Pade approxi- —AtB mations R (AtB*„0) are a rational matrix approximation for e~ M3 p,q M3 defined by -1 EP,a(wV V 4(4t W [dp,q(W;IW ] e-AtB^ + 0 ( At^)r (ll+5) where 1 - I k np,q.(AtIitp) = £ (P + q " + - k)'-3 P _1 v dp ,q ^ tBMp) = £ (p '+ q. - k)'[(p + q.)'ki (p - k)l] (At^.p) k=0 r = p + q + 1 Now from Equations ll+3, ll+l+, and ll+5 we. see that the following relation ships hold R = f; 0,1 Mp - V + <>(w)‘ (1W) Rx,o “ 4 = V + 0(lst)2 ri ,i = 4 2 ” v + o ( w ) 3 where the choice of 0 corresponds to the forward difference, backward difference, and Crank-Nicholson schemes, respectively. The notation in- 0 dicates that each term of the matrix F^p can be expressed as the sum of the corresponding term in the matrix E ^ plus a term of order of magnitude At** . Equation 11+6. implies that 80 Sm b - - 0(it)2 ^ B - FSb = 0<4t)2 {lkl) ^ B - FU» = 0<4t>3 Now (\ - where g^a is the fundamental matrix (0^,0^)^ . Thus from Equation lit 7 and lit 8 | ^(xjvAt^v - 1 ) At) - E^t,:L(x;vAt,(v - l)At)|| = C^At2 I |Eh(?;vAt, (v - 1)At) - E^ ’°(x;vAt,(v - 1)At) | | = CgAt2 | lE^xjvAt.iv - 1)At) - E^t,2 (x;vAt,(v - l)At)|| = C^ t 3 (1U9) Based on the previous results, the temporal error estimate theorem can he introduced. We will prove the temporal error estimate theorem only for the forward difference approximation (0 = l). Error estimates for other temporal operators can he derived in similar fashion. Theorem k.3. Let U(x,nAt) he the solution to the semidiscrete- Galerkin equation Equation 105 at time t = nAt and if'(x) he the solution to the forward differenced Galerkin approximation, Equation 107 with 0 = 1 , at time t = nAt . In addition let Equation ikO, 1^2, and lk9 hold. Then the temporal approximation error is I|T(x,nAt)||m = ||u(x,nAt) - ^(x)||m < CgAt2 ||u°(x)||m (150) Proof: Using the semigroup property, Equations 138 and 139 81 ||x(x,nAt)||m = ||u(x,nAt) - Un(x)||m = | | (E^xjnAt.O) - E^t,;L(x;nAt,0) )U°(x) | |m = || £ E^t,1(x;nAt,vAt)[E^t,1(x;vAt,(v - l)At) V=1 - Eh(x,vAt,(v - l)At)]Eh(x;(v - l)At,0)U^(x)||m 1 X ||E^t,1(x;nAt,vAt) | | | |E^t,:L(x;vAt,(v l)At) v=l - Eh(x,vAt,(v - l)At) | | | ^(xjiv.- l)At,0)U°(x)| |m And using Equations ihO 9 l k 2 9 and 1^-9 ||x(x,nAt)||m < | CllC54t2C3 ||0°(x)||m £ C j U t ) 2 ||U°(*)||m 110. The total error estimate of the approximation is given in the following theorem. Theorem U.U. If the components of the error of the approximation are given by Equations 109, 110, and 111, then ||e(x,nAt)||m £ C2hk+1_m|u(x,nAt)|k+1 + CgAt2 ||U°(x)||m (l5l) Proof: From Equation 112 ||e(x;nAt)||m £ ||a(x,nAt)||m + ||x(x,nAt)||m But introducing Equations 135 and 150 ||e(x,nAt)||m £ C2hk+1_m|u(x,nAt)|k+1 + CgAt2 ||u°(x)||m 82 Error Estimates for the Diffusion Equation Using the Energy Method 111. We often find it mathematically convenient and physically appealing to seek an error estimate that indicates the error in energy of an -approximation. This estimate of the error in energy of the ap proximation requires the definition of an appropriate energy norm. For the diffusion problem, the most natural energy norm can be constructed using the bilinear form a(u,v) connected with the operator A in Equation 101. The bilinear form a(u,v,) satisfies the m-ellipticity condition (Equation 102). Thus effectively, an inner product space is introduced in association with the operator A and the inner- product a (u,v) . The norm associated with the inner-product is called the energy norm, u = ya(u,u) u e H. ( 152) the properties of which depend intrinsically on the operator A . The energy space H is topologically equivalent to the Sobolev space Hm (Q) ; i.e., positive constants exist such that 0 a n d u Y-, | |u| | ( 1 5 3 ) 'm ullA i 1 1 1 1 'm In the subsequent developments we find it convenient to replace the energy norm || • ||^ by the Sobolev norm || • ||^ using Equation 153 realizing that || • I Im is fully equivalent to | | • | |^ in the energy sense. 112. Now suppose u|n is the exact solution to Equation 10k evaluated at time point t = nAt and Un is the solution to the finite difference-Galerkin equation (107) at time point t = nAt . We seek an estimate in energy for the difference between u|n and U31 . In this section we consider only the case in which the temporal operator is the forward difference operator in time. Thus we set 6 = 1 in Equation 107- Now let W11 be an arbitrary function in the subspace . 83 Then we define the error components .n e n = u In U W U En = u ln - W11 n du r | e = T7"r “ o u n (15*0 dt n t 1 We can now cite a theorem which specifies the behavior of the error. Theorem $.1. Let en , En , En , and z1 be the error components de fined in Equation 15^ • Then a(En,En) = -(6ten,En^ - a(En,En) - (en,En)0 (155) Proof: Subtracting Equation 107 with 0 = 1 from Equation 10U evalu ated at t = nAt ( i t L - + a(uln ■ljn,v) = 0 v £ V n) (156) where -^| denotes evaluated at time point t = nAt . Rewriting ot n dt Equation 156 (fl„ - Vl" * V ” - * a(u|n - + w11 - \Jn,Y) = 0 V e Sh(fi) or introducing Equation l$k (eU + |Sten ,V^ + a(En En ,V) = e (15T) V h Now setting V = En and using the bilinearity of (*,*)q and a(*,*), we obtain Equation 155- 81+ 113. The error component En can he estimated using the follow ing theorem. Theorem 5.2. If the error components en , En , En , and en are re related by Equation 155, then there exist positive constants yQ , ^l 5 y^ , and such that y (158) Proof: The last term on the right hand side of Equation 155 can be estimated using the Schwarz inequality for the inner-product and the embedding result ||En || £ c||En | . There exist positive con stants and such that n,En>0 ± Mil l En'lo liEnHo i H2 l l ^ l o l l E X (159) Then introducing Equations 10^ and 159 into Equation 155 En M0 (160) (E“ ,En ) < -(« t en,En) + y 11 ||1 ||MmM ||En|| 11 m + 2M||en|I 1101 ||En||1 'm Now to estimate (6 en,En) set V = En in Equation 157 \ t /Q /V n rn\ /rn rnv ^ /_n rnv , , nrnx - ^ e ,E J = a(E ,E ) + a(E ,E ) + (e E ) (161) Introducing Equations 152, 153, and 159 into Equation l6l (162) \ Now introducing Equation 162 into l60 and using the m-elliptic property of a(•,•) (Equation 109), we obtain the energy estimate. 85 The result in Equation 158 follows by dividing by ||En || . Ilk. Then the estimate of the approximation error e11 is con tained in the following theorem. Theorem 5.3» If the error components en , En , En , and e11 are defined by Equation 15*+ and En satisfies Equation 158, then there exist positive constants , M , Mg , and such that (163) and k+l-m u (16^) k+1 3 where Proof: The first result follows from Equation 15*+ and 158 . Equation 158 implies that 0 But from Equation 15*+ |En + r < (1 + M 3)I |En | 'm lm 'm 1 m — 'm + (165) The second result can be obtained by expanding u in a Taylor's series about t = nAt , introducing the result into Equation 15*+» and taking the L2 norm. | | enM < cAt2 |||u |M 3 (166) Then introducing Equations 97 and 166 into Equation l655 we obtain Equa tion l6k. 86 Error Estimates for the Diffusion Equation Using Lp Methods 115. The derivation of error estimates for finite element- Galerkin models for the diffusion equation follows very naturally from the interpolation theory developed in section 2. The key point in the derivation of the error estimate is establishing a bound on the approximation error for all time points based on the pointwise error estimate in time. Traditionally this has been carried out through a discrete version of Gronwall!s inequality (Lee^J). The method used here to derive error estimates for the finite element-Galerkin solution circumvents the complex arguments involved in using the discrete Gronwall's inequality. For simplicity, the finite element-Galerkin solution will be defined as piecewise linear between discretization points in time. Then we identify the nodal values in time, effectively, through a collocation procedure at the temporal node points. Thus we embed the approximate solution in the temporal, as well as the spatial, variables in a subspace of the original solution space. This procedure leads to the use of the previously developed interpolation results in time, as well as space. 116. We consider the solution of the weak parabolic problem (Equation IOU) with f(t) = 0 . The exact solution to this problem has Se been characterized by Goldstein. u e ch o .T jlA n )] (167) However if the coefficients A n in Equation 100 are such that 00 Ofg A _sC (2 ) and Dy(A 0)eL (2) for all a , 3 » and y , then u e C°°[0,T;Hm (u)] (168) We assume in this section that the coefficients A ag0 are constants so that Equation l68 holds. However, this choice is not basic to the method of derivation of error estimates to be introduced here. In fact, 87 the methods used here are valid as long as the regularity of the exact solution u is such that e L2 [0 ,T ;lf(a )] 9t •117. Let us define the space PCT*"[0 ,T;S^(i2) ] . If PchOjTjS (i2)] is the space of functions which are continuous with piecewise continuous derivatives between the points of the partition P of [0,T] introduced in section 3, then PC"^[0] is the subspace of PC1 [0 ,T;S (ft)] of functions which are piecewise linear.. And S (ft) P TT1 is the finite dimensional subspace of H (ft) described in section 2. We seek an approximate Galerkin solution Un to Equation 10U using the finite difference-Galerkin approximation (Equation 107) with 0 = 0 (this corresponds to the forward difference in time case). The solution of this problem is fully equivalent to the solution of the following problem. Find a U e PC1 [0,T;Sh (n)] such that ^ M E L 5v) ^ ^ + a(u(t),V)t=nAt = 0 ¥ V e Pcho.TjS^n) ] n “ 0 j ^ P (u (-,o ),v )0 = (u0 ,v)Q (169) where the notation indicates that a collocation is performed at time points t = nAt , n = 0,...,R . In the subsequent derivation of error estimates we will use the problem statement as in Equation 169 rather than in Equation 107 with 0 = 0 since the two problems are equivalent and the use of Equation l69 allows us to introduce the interpolation theory of section 2 and thus to avoid certain mathematical difficulties. 118. Now let W(t) be that element of PchOjTiS^ft) ] which interpolates u in the sense of Equation 87, and let Un = U(nAt) . We can define the error components of the approximation by 88 e = u ( t ) - U ( t ) E 1 = W(t) - U(t) e2 = W(t) - u11 E = u ( t ) - W ( t ) (170) Then the following theorem describes the behavior of the error. Theorem 6.1. If u(t) is the exact solution to Equation 10^, U(t) is the approximate solution to Equation 10k defined by Equation 1 6 9 , W(t) is the interpolant of u(t) defined by Equation 8 7 , and the error components are defined by Equation 170, then \ nAt <_ t < (n + l)At (171) Proof : Substracting Equation 169 from Equation 10^ - a(u(t),V(t))t nAt = ¥ V e PC1 [0,T;Sh (fi)] (172 ) But for nAt <_ t e n + l(At) . Thus ¥ V e PC1 [0,T;Sh (S2)] (173) Rewriting Equation 173 89 M i l _ Mi l + Mi l _ Mi l jV(t)) + a(u(t) - w(t) + w(t) 9t 9t 3t 9t / n - l^.vit)) = 0 ¥• V e PC1 [0,T;Sh (fi)] (17M Then introducing Equation 170 into Equation 17^- and setting V - E ) (175) (ir > Ei ) + a(E2’Ei> = - ( f - Ei) - a The result in Equation 171 follows directly from Equation 175 by split ting the second term on the right hand side. 119- The following theorem describes the variation of with time. Theorem 6.2. Let the hypotheses of Theorem 6.1 hold, then 2 1 i_| I II2 < _ (i* E ) + "l I, M t l ( t . „ 4t)||2 n4t < (n + l)At (176) Proof: - E n = U(t) - l/1 = (t = nAt) for nAt <_ t < (n + l) t ------2 1 a t Thus using the Cauchy-Schwarz inequality (Equation 103) a(E2 - E1,E1 ) = afSiii (t . nit) 3t > 0 V | P- 3U(t) m 1 - (t - nAt) | nAt <_ ' IT H 1 3t iji 1 —11 OJ 1 — V 1 2 Now using the elementary inequality ab 1 ea e > 0 + ( i > : (with a = nAt)||m , b = IIEJ and c n^ < * - L’ 2y0 2 |3U(t) a(E2 - E ^ E p < yi I ■ (t - nAt) nAt <_ t -2y0II at Nm + vol lEi 11! < (n + l)At (177) 90 Now by the definition of the interpolation operation, Equation 87, used to define W(t )ePC^[0,T;S (ft)] and the best approximation property of 22 ^ Hilbert spaces (Sard ), we have that (E,V)q = 0 ¥■ V e PC1 [0,T;Sh (i2)] (178) But by using Equation 101 and the symmetry of the bilinear form a(«,#) a(E,V) = a(V,E) * V e P c h o . T ^ f t ) ] = (AV,E)q (179) Now for the moment we suppress the dependence on time. If V e PC^[0,T;S^(Q)] , then for any particular time point, V e . Now contains all polynomials of degree less than k , and A is the operator of order 2M introduced in Equation 100. Thus A maps S (ft) into . This implies that if V e PC^tOjTjS^ifi)] , then AV e PC^[0,T;S (ft)] . Thus introducing Equation 1 7 8 into Equation 179 and setting V = E^ a(E,E1 ) = (aE ^ eJ = ^E,AE1^ = 0 (l80) Now introducing Equation 177 and Equation 180 into Equation 171 3U(t) 2 + a(E_L,E1 ) (t - nAt) 8t m + y I IE I 12 nAt < t < (n + 1 )At 0 1 'm — Now using the definition of the norm and the m-elliptic property of Equation 102 _ E ) + h_i 2 at11Ilf 11 Ilo + "ol 'm \3t* 1/ 2yQ + V~r nAt < t < (n + l)At (181) 91 2 Thus canceling 11 I |E, I I on each side of Equation l8l, we obtain the 0 1 m result in Equation l j 6 . 120. The following theorem gives an estimate for E^ at the dis cretization points in time. Theorem 6.3. Let the hypothesis of Theorem 6.1 hold, then IIL I Il-( l2) = |Ei (M4t) I lo ± S [ I IE1(0> I lo + I W Il2(l2) + 4t3/2||Mtl|| ([im)] (l82) Proof: Integrating Equation 176 from t = nAt to t = (n + l)At ~(n+l)At ||E1 (n + l)At| |2 - | |E1(nAt)||2 < - 2 J ' ( H ’El)dt nAt (n+1)At 9U(t) (t - nAt)jI2 dt (183) / 3t 'm nAt Now using the Schwarz inequality ,(n+l )At -(n+l)At >9U(t) (t - < / il3U(t)ll2 ( t ! 3t I» - J 11 at 11m nAt1 nAt f(n+1)At (n+1)At / Mau(t)M2 nAt) dt < «I (t ) 11 3t 1 nAt nAt 1 n+1)At AtJ / M au(t) - nAt) dt = | |2 dt (1 8 M 6 / 11 at 1 'm nAt 92 Introducing Equation 183 into Equation l8U and summing from n = 0 to n = M M M M ^(n+l)At dt i HE1((n + l)AtNo - Z MEl(nAt)M0 - - 2Z j ( l f ’ El ) n=0 n=0 n=0 nAt 2 M Jn+l)At 1 3 3U(t)||2 + At3 dt dt ' 'm % n=0u nAt Or MAt 9E [E (MA-t) | ¡1 11^(0) | |q <. 2^ at’Ei 0 'at 2 MAt (185) Then integrating the first term on the right hand side by parts and 1 2 2 using the inequality (a,b) l_]^lla ll + n | |b | | (n > 0) MAt l|E1(MAt)||2 - ||E1(0 )|\q + dt MAt + 93 3E, In addition But E(t) e PC±[0,T;Sh («)] , thus at 11o -< C, l I11 IE, l1'o * au(t)i <_ C | | 0 | | for all te[0,MAt] . These results imply dt 1 1 m — ~2 11 3t 11 m that |El(HAt)| < [ | ¡5^0)11^ ♦ |E|\h2 ♦ ^ ^ | \ £ g l 2 2' T + 2n C1 J | (186) 'l2(h“)- I Ei I I o dt But the classical Gronwall's inequality (Bellman^) implies that if |x(t)| <_ a + /^|x(s)|c ds where a and C are positive constants, 0 Ct then x(t) < ae . Thus lEi u2C + X 2 3, |lu(.t) i 12 -i 2nC t 6y At 11 at 1lL2 (Hm )J 6 1 or lEqiMAt) I l0 ± c 3 [ I 1 ^ ( 0 ) | |Q + | |E| 1 ^ ^ ) 3/2 3 u ( t ) + At (187) 3t The result, Equation 182, follows by taking the supremum of Equation 1 8 7 for all integers M with 0 <_ M <_ R . Note that PC^"[0,T;S (ft)] is __p -*-1 a space of piecewise linear functions. Thus since E^ e PC [0,T;S (ft)] , E attains its maximum at one of the discretization points in time. Thus the supremum of the E^ at the discretization points in time is exactly the L^ norm. 121. The approximation error can then be established through the following theorem. Theorem 6.U. Let the hypotheses of theorem 6.1 hold, then 3/2, 3u(t ) i(0) + At ] (188) L2(L2) w 3t Proof: Using Equation 170 l'e' 'l J l J " I 'E l 'l J l J + I lEi' 'L0 (L0 ) 2 2 ' 2 2 2 2 (189) - 1 |E* + t ||Ei ,Il oo(l 2 ) Then introducing Equation 182 into Equation 189 l'llLJ L„) i c3||E1(0)| |0 + ( 1 + C3T)||E|| ( J 2 2 2 2 ' 3/2 I.3U(t) + C3At lL (Hm) l C u[||e(0)||0 11 at I * , .3/2, i 3U(t ) i | m -, + E 1 2( Lg) At II 3t IlLgiH1“)1 122. We can now use the interpolation results of section 2 to derive the final error estimate. Theorem 6.$. If the space PC1[0,T;S^(Q )] satisfies conditions (i), (ii), and (iii) in section 2, then k+11 lellL2(L2 ^ CJ l e(0)ll0 + C7hJ |u|lL (Ht+ l) + °84t I Ih2(l 2) + V t3/2Hf l2(h») (190) Proof: The interpolation error E is caused by an orthogonal projec tion Q in space and Q in time. The projection operator & x x » n 95 (Equation 87) is the composition Si _ QxQt Then using the triangle inequality and the fact that IIQxII 5 . 1 I lEl Il2(l 2) = Nu - Siul Il2(l2) = ||u - Qxu + Qxu - QxQtu|lL2(L2) < I|u - Qxu IlL2(L2) + I|Qx (u - Qtu)IlL2(L2) — I |u - V l Il2(L2) + 1 |Qx! 1 I |u ■ QtUl Il2(L2) (191) QxUN l2(L2) + N u - VH l2(L2) But from the interpolation result, Equation 97 il"- v Hl2(l2) i I|u - Qtu|Il2(L2) - c54t Il" lI h2(L2) (192) Then introducing Equation 191 and Equation 192 into Equation l88, we obtain the estimate in Equation 190. 96 PART VII: CONCLUSIONS AND RECOMMENDATIONS 123• A number of FE and FD schemes were investigated for the problem of one-dimensional consolidation governed by a parabolic equa tion. Both quantitative (parametric) and mathematical analyses were performed. These analyses permitted conclusions regarding tradeoffs between accuracy, stability, and cost for various schemes. The results can be of value to the user interested in choosing an appropriate scheme for his needs. 12k. A study of the concept of finite elements in the time domain and examination of numerical properties of time integration schemes for general dynamical systems were initiated. Only partial results for the former towards study of a single-degree-of-freedom dynamical system and formulation of some time-dependent problems were obtained and are included herein. Additional investigations will be needed for completion of this aspect and the topic of numerical con vergence and stability of general dynamical systems. 125- A detailed mathematical analysis for error estimates of FE and FD approximations of diffusion equations is presented. Three dif ferent procedures— semigroup theoretic estimates, energy approaches, and schemes— are used for the error analysis. 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A., "General Lagrange and Hermite Interpolation in Rn with Applications to Finite Element Methods," Archives for Rational Mechanics and Analysis, 46, 1972, pp 177-199* 51. Friedman, A., Partial Differential Equations, Holt, Rhinehart, and Winston, 1969. 52. Schultz, M., Spline Analysis, Prentice-Hall, 1973* 53. Lees, M., "A Priori Estimates for the Solution of Difference Ap proximations to Parabolic Partial Differential Equations," Duke Mathematical Journal, 27, i960, pp 297-311* 54. Goldstein, J., Semigroups of Operators and Abstract Cauchy Problems, Tulane University, 1970. 55. Bellman, R., Stability Theory of Differential Equations, McGraw- Hill, 1952. 101 Table 1 Relative Errors and Computational Times Number Relative Error of Nodes At Linear Cubic Error u* - u n 0.2 1 0.90 0.1+jif 1 : 3-00 21 0.24Ì 1 : 0.90 O.bJl) 1 • 3.00 U* - u 11 1 : 1.02 21 1 • 1.0 1 Total Time 11 1 3.O 21 1 U.O 26 1 k.O 2 NB (Measure of Time for Solution of Equation Sets) 1 8 Table 2 Properties of Layered Systems for Load Increment 20-U0 psi Two-Layer System Three-Layer System c c Y k Y k 2 Y 2 Y Material ft /year ft/year Material ft /year ft/year Top Layer Kaolinite 75-5 0.103 Derwent + 1.25$ 7 .7 1 0.0123 bentonite Middle Layer Derwent 12.U5 0 .0 179 Bottom Layer Derwent + 1.25$ 7.71 0.0123 Kaolinite 75-5 0 .103 bentonite T ab le 3 Closed Form Solution, Equation 67 ____ Time_____ Displacement, u Velocity, û 0.000000000E+00 0 . 000000000E+00 6.283180000E-01 1.909826937E-01 I. 256636OOOE+OO 6.909819961E-01 1 .88U95^000E+00 1 .3 0 9 0 1 5 1+90E+00 2 .513272000E+00 1 .8 0 9 0 1 5 7 1+7E+00 3 .1^1590000E+00 2 .00000000ÛE+00 3 .T 699080OOE+OO I. 8O918866E+OO Ì+.3982260OOE+OO 1.309020528E+00 5 .0 2 6 5 1+1+000E+00 6.9098701+36E-01 5.65Ì+862000E+00 1.909858132E-01 6.293180000E+00 1 . U55191523E-II FEM Solution for NEL = 0.000000000E+00 0.000000000E+00 0.000000000E+00 I.570795OOOE+OO 1.000218990E+00 1.022325958E+00 3 .1^1590000E+00 2 .OOOO818U9E+OO -1.51+6lU0993E-ll Ì+.712385000E+00 I.OOO21899OE+OO -I.O22325958E+OO 6.283180000E+00 0.000000000E+00 0.000000000E+00 Table k FEM Solution for NEL = 10 _____ Time______Displacement, u Velocity, ù 0.000000000E+00 O.OOOOOOOOOE+OO O.OOOOOOOOOE+OO 6.283180000E-01 1.909853126E-01 5.88306Uo 1+5E-01 1.256630000E+00 6.90986001+5E-01 9.518961+021E-01 1.88^95^000E+00 1.309019195E+00 9.5l89^8it73E-01 2.513272000E+00 1.809017962E+00 5.88302915^E-01 3.1^1590000E+00 2.000000UO3E+00 -2 .IO88908UOE-IO 3.769908000E+00 1.809017961E+00 -5.883029156E-01 k .398226000E+00 1.309019195E+00 -9-5l891+8U73E-01 5 • 02ô5^000E+00 6.90986001+3E-01 -9.5l896i+020E-01 5.65^862000E+00 I.909853125E-OI -5.883061+01+3E-01 6.283180000E+00 O.OOOOOOOOOE+OO O.OOOOOOOOOE+OO Table 5 FEM Solution for NEL = 20 _____ Time______Displacement, u Velocity, ú O.OOOOOOOOOE+OO 0.000000000E+00 O.OOOOOOOOOE+OO 3.IU159OOOOE-OI !+. 89^ 156772-02 3 .09037^157E-01 6.28318OOOOE-OI 1 .9098^19732-01 5.878222U78E-OI 9 .i+2i+770000E-01 k. 1221521832-OI 8 .09066715^2-01 I.256636OOOE+OO 6.9098U515IE-OI 9.5111^175IE-OI I.570795OOOE+OO l . 0000013Í+92+00 I.OOOO6OIOIE+OO 1 .88it951*000E+00 I.3O9OI8O52E+OO 9.511133^3^2-01 2 .I99II 3OOOE+OO 1 .5 8 7 7 8 5966E+OO 8 .0906511+26E-01 2 .513272OOOE+OO I.809OI7392E+OO 5.878201U37E-OI 2 .827it31000E+00 1 .95105669^2+00 3 .O90353II 6E-OI 3 •1U159000OE+OO 2 .000000100E+00 1 .3^ 858W7E-09 3 .i*557li9000E+00 I.95IO566952+OO -3.090353087E-OI 3.7699080OOE+OO 1 .80901739^2+00 -5.878201IH2E-OI U.08U067OOOE+OO I.587785968+OO -8 .0906511t08E-01 k .3982260OOE+OO I. 309OI8055E+OO -9 .5111331+25E-01 ^.712385OOOE+OO 1.000001352E+00 -1 .OOOO6OI02E+00 5 .02651+1+000E+00 6.90981j5181E-01 -9 .51111+1762E-01 5 .3H0803OOOE+OO U.122162208E-OI -8.O90667173E-OI 5 .65Ì+862000E+00 1.9098Ì+1991E-01 -5.878222503E-OI 5.96902100OE+OO I+.89UI+15767E-O2 -3 .09037^l82E-01 6.28318OOOOE+OO 0.000000000E+00 O.OOOOOOOOOE+OO In accordance with ER 70-2-3, paragraph 6c(l)(b), dated 15 February 1973* a facsimile catalog card in Library of Congress format is reproduced below. Desai, Chandrakant S Evaluation and analyses of some finite element and finite difference procedures for time-dependent problems, by C. S. Desai, J. T. Oden, ¿andj L. D. Johnson. Vicksburg, U. S. Army Engineer Waterways Experiment Station, 1975. vi, c107j p. illus. 27 cm. (U. S. Waterways Experiment Station. Miscellaneous paper S-75-7) Prepared for Office, Chief of Engineers, U. S. Army, Washington, D. C. Bibliography: p.98-101. 1. Finite difference method. 2. Finite element method. 3. Soil mechanics. 4. Time dependence. I. Oden, John Tinsley, joint author. II. Johnson, Lawrence D., joint author. III. U. S. Army. Corps of Engineers. (Series: U. S. Waterways Experiment Station, Vicksburg, Miss. Miscellaneous paper S-75-7) TA7.W34m no.S-75-7