II II II II II I - Waste Isolation Pilot Plant - U.S. Department of Energy

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iteration matrix which relates the velocities and displacements at step n+ 1 to those at step n, the expression for the damping parameter, 6, is found to be s = 1- (4cooml)/ ((OO+CO])2 . (3.7.6) The allowable range on 6 is (O,1). A stability analysis on this set of explicit equations produces a critical pseudo-time step given by ATC=2/@o+col)@ . (3.7.7) If the problem is linear so O. and 01 are fixed, then the number of time steps, N, required to reduce the vibration amplitude by a factor often is N= 1.15 (co1/coo). (3.7.8) From this equation, it is seen that any effort to reduce the ratio o 110)0speeds convergence. From the linear problem and a uniform mesh of dimension Ax, the maximum frequency Q 1 is given by C01=2CIAX=21AT . (3.7.9) In this expression, c is the dilatational wave speed given by c=(k+2p)/r (3.7.10) and r is the pseudo-density used for the computation of the fictitious mass. If we substitute the quantity 2/A~ fom 1 and remember that 01>> coo, then the expression for the damping parameter becomes 8=1-200AT . (3.7.11) The fimdamental ilequency O. is continuously estimated using an approximate value found using the Rayleigh Quotient. At each iteration i in the dynamic relaxation scheme, a new estimate (~o)i is computed as “o,=- (3.7.12) ‘ where K is a diagonal stiffness matrix whose jti component is computed from f.jmT – f;-yT K.i=l (3.7.13) ATU:-l “ With each estimate of the fundamental frequency, a new value of the damping is computed. This has the virtue that the lowest active mode will be found in the event that the fimdamental mode is not participating (Underwood, 1983). 28
3.8 Convergence Measures When an iterative method, such as dynamic relaxation, is used to solve for static equilibrium, some criterion must be used to determine when the estimated solution is sufficient] y close to the actual solution. Convergence of the equilibrium iteration process is achieved when a measure of the problem force imbalance reaches a value less than or equal to a user-supplied error tolerance. The force imbalance is the sum of the external and internal nodal forces which at equilibrium should sum to zero. In SANTOS, two different convergence error measures are available to the analyst. The first error measure is based on satisfying the following inequality: Rj — < TOL Fn (3.8.1) where II0II denotes the Lz norm of a vector, Rj is the residual or imbalance force vector at iteration j, and Fn is the external force vector at step n which is composed of applied tractions, body forces (gravity forces), thermal forces, and the reactions at nodes where zero displacement boundary conditions are applied. Equation 3.8.1 is a measure of how close the problem is to a state of equilibrium. The quantity TOL is input by the analyst as a means of identifying the relative imbalance the analyst is willing to accept in the solution. In SANTOS, TOL is set by default to a value of 0.5 percent. This error is called the GLOBAL CONVERGENCE measure and is the default error measure. The second error measure implemented in SANTOS is based on satisfying the error tolerance on a node-by-node basis. This error measure is called the LOCAL CONVERGENCE measure. The rationale for this criterion is that what is an acceptable force imbalance in one portion of the problem may be unacceptable at another location. For example, further reduction of a set of force residuals acting in a region of the problem where the elements are large and stiff may produce only a small change in the element stresses in this region. If, however, the same set of force residuals was present at a different location where the element sizes were much smaller and the material was much more flexible, further reduction of the residuals could produce a large change in the element stresses. To address these concerns, the LOCAL CONVERGENCE error measure is included as an option. The error measure for each component i and iteration j of the residual force vector is defined as: (3.8.2) where R ~ is the residual or imbalance force, F; is the external force, f j is the internal force, and f ~in is the minimum internal force in an element produced by a reference hydrostatic stress state specified by the analyst. The minimum internal force is introduced to ensure that the denominator is never zero and to prevent elements with negligible stresses from controlling the convergence of the problem. The internal force contribution is summed over 29
Page 1 and 2: SANDIA REPORT SAND90-0543 l UC-721
Page 3 and 4: SAND90-0543 Unlimited Release Print
Page 5 and 6: CONTENTS 1.0 IN~ODUCTION . . . . .
Page 7 and 8: Figures 2.1.1 2.2.1 2.2.2 3.1.1 3.7
Page 9 and 10: 1.0 INTRODUCTION SANTOS is a finite
Page 11 and 12: 2.0 GOVERNING EQUATIONS In this cha
Page 13 and 14: Using the left decomposition of Equ
Page 15 and 16: 1.5 1.0 0.5 a 0.0 i% \ +7797 / \ &
Page 17 and 18: 3.0 NUMERICAL FORMULATION In this c
Page 19 and 20: m’ I 2$=2 y 4 L x I 1 2 1- 1 I I
Page 21 and 22: Consequently, the gradient operator
Page 23 and 24: We are now faced with a three-dimen
Page 25 and 26: We now see that the internal force
Page 27 and 28: ‘t+At = QAt % (3.3.2) For a const
Page 29 and 30: If the strain increments are insign
Page 31 and 32: The mean coordinates Z, correspond
Page 33 and 34: for arbitrary Ui]. Using Equation (
Page 35: TRI-6348-6-O Figure 3.7.1. A model
Page 39 and 40: 4.0 CONSTITUTIVE MODELS One of the
Page 41 and 42: where y is a column vector containi
Page 43 and 44: ~=~y —=Kdkk (4.3.3) where K is th
Page 45 and 46: Q TRI-6348-7-O Figure 4.5.1. Yield
Page 47 and 48: @r=R . (4.5.16) Because S is deviat
Page 49 and 50: and (4.5.27) Combining Equations (4
Page 51 and 52: TRI-6348-1o-o Figure 4.5.4. Effect
Page 53 and 54: The subscripts n and n+ 1 refer to
Page 55 and 56: The elastic plastic material model
Page 57 and 58: (a) a. P, Compression (b) 16 a. ‘
Page 59 and 60: The mean volumetric strain is updat
Page 61 and 62: ~=$s:s . (4.6.9) J The yield condit
Page 63 and 64: in Section 4.6, use such decomposit
Page 65 and 66: not amenable to vectorization. Vect
Page 67 and 68: We make use of the following identi
Page 69 and 70: law secondary creep model is reques
Page 71 and 72: PROP(3) - Modulus function identifi
Page 73 and 74: where K @ () n+l is the O;:l =3K @n
Page 75 and 76: 2 11 F=exp-1-~ 6 for q>&; . (4.12.8
Page 77 and 78: Accuracy of the method is assured i
Page 79 and 80: The volumetric part of the model ca
Page 81 and 82: for the bulk response, where G(t) a
Page 83 and 84: then G(t,@) = G(g) . (4.14.9) This
Page 85 and 86: PROP(7) - Third Short Time Shear Mo
Page 87 and 88:
5.0 CONTACT SURFACES Many structure
Page 89 and 90:
5.2 Application Phase The operation
Page 91 and 92:
6.0 LOADS AND BOUNDARY CONDITIONS S
Page 93 and 94:
TRI-6348-43-O Figure 6.2.1. Definit
Page 95 and 96:
diagonal mass matrix routines with
Page 97 and 98:
Biffle, J. H., and M.L. Blanford. 1
Page 99 and 100:
Schreyer,H.L., R.F. Kulak, and J.M.
Page 101 and 102:
APPENDIX A: SANTOS Users Manual A-1
Page 103 and 104:
SANTOS Users Manual Listed below, i
Page 105 and 106:
tol ... tolerance value used for co
Page 107 and 108:
See the theory section for definiti
Page 109 and 110:
omega ... specified angular velocit
Page 111 and 112:
dence of the pressure scale factor
Page 113 and 114:
TWO MU(2P) BULK MODULUS (K) AO Al A
Page 115 and 116:
RN3 (exponent of workhardening and
Page 117 and 118:
APPENDIX B: User Subroutines B-1 ,.
Page 119 and 120:
‘ser Subroutines SANTOS allows th
Page 121 and 122:
600 CONTINUE END IF 1000 CONTINUE R
Page 123 and 124:
APPENDIX C: Printed Output Descript
Page 125 and 126:
Printed Output Description The SANT
Page 127 and 128:
FUNCTION DEFI NIT IONS FUNCTION ID
Page 129 and 130:
1 SANTOS, VERSION SANTOS 2.0 ,RUN O
Page 131 and 132:
APPENDIX D: Adding a New Constituti
Page 133 and 134:
Adding A New Constitutive Model to
Page 135 and 136:
APPENDIX E: Verification and Sample
Page 137 and 138:
Verification and Sample Problems Sa
Page 139 and 140:
To obtain a solution to the gravity
Page 141 and 142:
Figure E-5. Deformed Shape of the B
Page 143 and 144:
P= () 4:(1-V) 1-: pc3 +2(1-m)lllpc
Page 145 and 146:
L < 1 t i I I t I , I 1 3 3 1 1 I i
Page 147 and 148:
Side Set ID for the Rigid Surface C
Page 149 and 150:
TITLE UPSETTING OF A CYLINDRICAL BI
Page 151 and 152:
Table 1: M-D Argillaceous Salt Cree
Page 153 and 154:
4.0 3.5 3.0 1.0 0.5 0.0 I I I 1 1 1
Page 155 and 156:
WIPP UC721 - DISTRIBUTIONLIST SAND9
Page 157 and 158:
Sue B. Clark University of Georgia
Page 159 and 160:
Mr. Steven Crouch GeoLogic Research
Page 161:
[:1 . 1 .i i- ) r
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