Belytschko T. - Introduction (779635), страница 59
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Thus, viewing the solution of the residual equations asa minimization problem is helpful in many cases.6.3.2b. α -METHOD EQUATIONSThe α -method, also known as Hilber-Hughes-Taylor (HHT) method [??],was introduced to improve numerical dissipation for high frequencies in theNewmark-β method. The Newmark- β formulas, Eqs. (6.3.4) - (6.3.7), remainthe same, wheras the time-discrete equation of motion is modified as follows (cfEq. 6.3.1)()()(0 = r dn +1, t n +1 = sDMan +1 − f ext dn +α , t n +1 + f int d n+ α)(6.6.1)wheredn + α = ( 1+α )d n+1αd n(6.6.2)It is noted that in the case of a linear analysis, the internal force vector becomes:()f int dn +α = Kdn + α = (1+α )Kdn+1 −αKdn(6.6.3)which is exactly the HHT method presented in [??].
Follow the α -methodstability analysis, unconditional stability is achieved by setting the followingparameters:(1− 2α )(1− α )2 1 α ∈ − , 0 ; γ =; and β = 3 246-15(6.6.4)T. Belytschko & B. Moran, Solution Methods, December 16, 1998If α = 0 , the trapezoidal rule is obtained.Remark: Although there is no stability analysis in the literature for a nonlinearsetting (i.e., with Eq. (6.6.1)), a linearized stability analysis will yield the samestability result as in Eq. (6.6.4).Following the same procedure given in Section 6.3.2, the discrete problemas stated in Eq. (6.3.10) is revised as:(n +1find d so that r( dn+1) = 0 as shown in Eqs.
(6.6.1) - (6.6.2), subject to)g dn +1 = 0 .In order to define the Jacobian matrices and incremental nodaldisplacements given in Eqs. (6.3.18) through (6.3.25), the following linearizeddisplacement equations are defined (cf. Eq. (6.3.21)):def˜dvn++1α = (1 +α )(dvn +1 + ∆d ) −αd n = dvn+ α + ∆d(6.6.5a)dvn++1α = (1 +α)d vn+1 −αdn(6.6.5b)whereandfor υ = 0∆˜d = (1+ α )∆da n0 +1 = 0, d n0 +1 = d n + ∆tυn +∆t2(1−2β )a2(6.6.5c)With the above definitions, the linearized Jacobian matrix equations becomes: (cf.Eq.
(6.3.17))r( dn+ αv)+∂r(d vn+α )∂d∆d +0 ( ∆d2 ) = 0(6.6.6)The Jacobian matrix or the effective tangent stiffness matrix (cf. Eq. (6.3.22)) canbe shown to beA=(∂r d n+αv∂d)=()(∂fint dnv +α∂f ext dnv +αsDM + (1+ α )− (1+ α )β∆t 2∂d∂d)(6.6.8)The rest of the formulation remains the same.6.3.5.Implementation of Newton Method. Flowcharts for implicitintegration and equilibrium solutions are given in Boxes 6.3 and 6.4.
Both thedynamic problem and the equilibrium problem are solved by time-stepping: theexternal loads and other conditions are described as functions of time, which isincremented over the range of interest. In equilibrium problems, the time is often6-16T. Belytschko & B. Moran, Solution Methods, December 16, 1998replaced by a monotonically increasing parameter.
Solutions of equilibriumprocesses obtained in this manner are called incremental solutions.The flowchart shows a procedure often called a full Newton algorithm, where theJacobian matrix is inverted in every iteration of the procedure. Many programsuse a modified Newton algorithm, in which the Jacobian is only triangulated atthe beginning of the iterations or intermittently during the iteration. For, example,in a modified Newton procedure the Jacobian may be triangulated only when theietrative procedure does not seem to be converging well. These modified schemesare faster but less robust.The flowcharts begin with the imposition of the intial conditions.
Theinitial conditions can be handled exactly as in explicit methods. The initialdisplacements are considered to be zero. The initial accelerations are computed asshown in steps 2 and 3.The displacements dn +1 for each time step are obtained by the iterativeNewton procedure. To begin the iterative procedure, a starting value of d isneeded; usually the solution from the preceding step is used.
The residual is thencalculated for this starting value. In an equilibrium solution, the residual dependsonly on the internal and external nodal forces. and is obtained in the module getf.This module, getf, is the same as in the explicit procedure, Box 6.1, except thatthe calculation of the stable time step is omitted, so it is not repeated. In transientimplicit solutions, the residuals also depend on the accelerations.Box 6.3Flowchart for Implicit Time Integration1. Initial conditions & initialization of parameters:set v 0, σ0 ; d0 = 0, n = 0,t = 0; compute M(2.
get f 0 = f d0 ,0)3. compute initial accelerations a n = M −1f n4. estimate next solution d : d = dn5. Newton iterations for time step n +1a. get f computes f d, t n +1n +1 ˜ n1˜nb. a n +1 = β∆t= v + γ∆tan+1 , see Eqs. (6.3.4 - 6.3.7)2 d−d , v(())c. r = Man+1 − fd. compute Jacobian A (d )e. modify A (d ) for essential boundary conditionsf. solve linear equations ∆d = A −1rg. d ← d +∆dh.
check error criterion; if not met, go to step 5a6. update displacements, counter and time: dn +1 = d, n ← n +1, t ←t +∆t7. check energy balance8. output, if simulation not complete, go to 36-17T. Belytschko & B. Moran, Solution Methods, December 16, 1998Box 6.4Flowchart for Equilibrium Solution1. Initial conditions and initialization: set u0 = 0; σ0 ; n = 0;2. Newton iterations for load increment n +1a. getf computes f d, t n +1 ; r = f d, t n +1()()b. compute A (d )c. modify A (d ) for essential boundary conditionsd. solve linear equations ∆d = A −1re.
d ← d +∆df. check error criterion; if not met, go to 2a3. update displacements, counter and time: dn +1 = d, n ← n +1, t ← t + ∆t4. output, if simulation not complete, go to 2The Jacobian matrix in this algorithm is then calculated based on the lateststate of the body. In some algorithms, the Jacobian for the last convergedsolution is used for all the iterations or the Jacobian is only recomputedintermittenly during the iterations; these are known as modified Newton methods.Simple essential boundary conditions, such as homogeneous displacementconditions, can be enforced by modifying the Jacobian matrix.
The equationcorresponding to the vanishing displacement component is either omitted orreplaced by a dummy equation that the component vanishes by zeroing thecooresponding row and column and putting a one on the diagonal of the Jacobian.For more complex algebraic constraints, Lagrange multipliers methods or penaltymethods are used: these are described in Section 6.?.6.3.6.Equilibrium Solutions Based on Stationary PotentialEnergy. In Chapter 4 we saw that when the system is conservative, i.e. whenthe stresses and external loads are derivable from a potential, then the equilibriumproblem can be posed as the determination of the stationary points of the energy.Such problems are called conservative. Stable equilibrium solutions correspondto local minima of the potential energy.Consequently, stable solutions for conservative problems can be found byminimization techniques.
The discrete problem is then: for any time t (the timeparametrizes the external load):min W (d, t) subject to g I (d) = 0 I =1 to nc(6.3.31)where g I (d) = 0 are nc discrete constraints on the system. These must be linearalgebraic constraints. If they involve differentials or integrals, they must beconverted to algebraic from by time discretization. Displacement boundaryconditions are often imposed as auxiliary consraints of this type.
Often theessential boundary conditions can be met by simply eliminating nodaldisplacements from the unknowns. If both linear stable and linear unstable6-18T. Belytschko & B. Moran, Solution Methods, December 16, 1998solutions are desired, then the the stationary points of W (d,t ) must be found. Thediscrete problem is thenfind d so that∂W (d)=− f = r = 0 subject to g I (d) = 0 I =1 to nc (6.3.32)∂dSolutions to these equations for loads which vary as a function of the parameter t ,which could be time but need not be, appear as branches (lines) in the space of thenodal displacement components. Some examples are given in Section 6.??.In the above we have indicated that the derivatives of the potential with respect tothe nodal displacements is the negative of the nodal forces, which are in turnequal to the residuals.
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