Belytschko T. - Introduction (779635), страница 10
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(2.3.16), and use thefinite element approximation of the test function (2.4.3), givingδWint ≡∑∫XbδuI f intI = δu, X PA0 dX =XaI∑ ∫Xbδu I N I , X PA0 dX(2.4.5)XaIFrom the above definition it follows thatXbf Iint = ∫X NI,X PA0 dX(2.4.6)awhich gives the expression for the internal nodal forces. It can be seen that the internalnodal forces are a discrete representation of the stresses in the material.
Thus they can beviewed as the nodal forces arising from the resistance of the solid to deformation.The external and nodal forces are developed similarly. The external nodal forces areobtained by using (2.4.4b) and (2.3.17) in conjunction with the test function:N(δW ext = ∑ δu I f Iext = ∫X δuρ 0 bA0 dX + δuA0 t xXbaIN= ∑ δuII{∫XbXa(00NI ρ 0 bA0 dX + N I A0 t x))ΓtΓt (2.4.7)where in the last step (2.4.3) has been used. The above giveXb(f Iext = ∫X ρ0 N I bA0 dX + NI A0 t xa0)(2.4.8)ΓtSince NI ( X J ) = δ IJ the last term contributes only to those nodes which are on theprescribed traction boundary.2-17T.
Belytschko, Chapter 2, December 16, 1998The inertial nodal forces are obtained from the inertial virtual work (2.4.4c) and(2.3.20):..δW inert = ∑δuI f Iinert = ∫X bδuρ 0 u A0dXX(2.4.9)aIUsing the finite element approximation for the test functions, Eq. (2.4.3), and the trialfunctions, Eq. (2.4.1) gives∑ δu I fIinert = ∑ δu I ∫X ab ρ 0 NI ∑ NJ A0 dXXIIu˙˙ J(2.4.10)JThe inertial nodal force is usually expressed as a product of a mass matrix and the nodalaccelerations. Therefore we define a mass matrix by∫XbMIJ = ρ 0 NI NJ A0 dXM=orXaXb∫Xρ0 NT NA0dX(2.4.11)aLetting ˙u˙ I ≡ aI the virtual inertial work isδW inert = ∑δuI fIinert = ∑ ∑ δuI M IJ aJ = δuT Ma, a = ˙u˙II(2.4.12)JThe definition of the inertial nodal forces then gives the following expressionf Iinert = ∑ MIJ aJorf inert = Ma(2.4.13)JNote that the mass matrix as given by Eq.
(2.4.11) will not change with time, so it needs tobe computed only at the beginning of the calculation. The mass matrix given by (2.4.11) iscalled the consistent mass matrix.Semidiscrete Equations. We now develop the semidiscrete equations, i.e. the finiteelement equations for the model. At this point we will also consider the effect of thedisplacement boundary conditions.
The displacement boundary conditions can be satisfiedby the trial and test functions function by lettingu1(t) = u1(t)andδu1 = 0(2.4.14)The trial function then meets Eq. (2.3.5). For the test function to meet the conditions ofEq. (2.3.6), it is necessary that δu1 = 0 , so the nodal values of the test function are notarbitrary at node 1. Our development here, as noted in the beginning, specifies node 1 asthe prescribed displacement boundary; this is done only for convenience of notation, and ina finite element model any node can be a prescribed displacement boundary node.We will now derive the discrete equations.
It should be noted that Eqs. (2.4.4a-c) aresimply definitions that are made for convenience, and do not constitute the discreteequations. Substituting the definitions (2.4.4a-c) into Eq. (2.3.21) gives2-18T. Belytschko, Chapter 2, December 16, 1998nN∑ δuI ( fIint − fIext + f Iinert ) = 0(2.4.15)I =1Since δuI is arbitrary at all nodes except the displacement boundary node, node 1, itfollows thatf Iin t − f Iext + f Iinert = 0, I = 2 to n N(2.4.16)Substituting (2.4.13) into (2.4.16) gives the discrete equations, which are known as theequations of motion:nN∑ MIJJ =1d 2 uJintext= 0, I = 2 to nN2 + fI − fIdt(2.4.17)The acceleration of node 1 is given in this model problem, since node 1 is a prescribeddisplacement node.
The acceleration of the prescribed displacement node can be obtainedfrom the prescribed nodal displacement by differentiating twice in time. Obviously, theprescribed displacement must be sufficiently smooth so that the second derivative can betaken; this requires it to be a C1 function of time. If the mass matrix is not diagonal, thenthe acceleration on the prescribed displacement node, node 1, will contribute to the Eq.(2.4.17). The finite element equations can then be written asnN∑ MIJJ= 2d 2 uJd 2 u1intext+f−f=M, I = 2 to nNIII1dt2dt 2(2.4.18)In matrix form the equations of motion can be written asMa = f ext – f intor(2.4.19)f = Ma, f = f ext – f intwhere the matrices have been truncated so that the equations correspond to Eq.
(2.4.17),i.e. M is a (nN −1) × n N matrix and the nodal forces are column matrices of order n N −1.The effects of any nonzero nodal prescribed displacements are assumed to have beenincorporated in the external nodal forces by lettingf Iext ← f Iext + M I1d 2u1dt 2(2.4.20)Thus, when the mass matrix is consistent, prescribed velocities make contributions tonodes which are not on the boundary. For a diagonal mass matrix, the accelerations ofprescribed displacement nodes have no effect on other nodes and the above modification ofthe external forces can be omitted.Equations (2.4.17) and (2.4.19) are two alternate forms of the semidiscretemomentum equation, which is called the equation of motion. These equations are calledsemidiscrete because they are discrete in space but continuous in time. Sometimes they are2-19T. Belytschko, Chapter 2, December 16, 1998called discrete equations, but they are only discrete in space.
The equations of motion aresystems of nN −1 second-order ordinary differential equations(ODE); the independentvariable is the time t. These equations can easily be remembered by the second form in(2.4.19), f = Ma, which is the well known Newton's second law of motion. The massmatrix in finite element discretizations is often not diagonal, so the equations of motiondiffer from Newton's second law in that a force at node I can generate accelerations at nodeJ if MIJ ≠ 0 .
However, in many cases a diagonal approximation to the mass matrix isused. In that case, the discrete equations of motion are identical to the Newton's equationsfor a system of particles interconnected by deformable elements. The force f I = f Iext − fIintis the net force on particle I. The negative sign appears on the internal nodal forces becausethese nodal forces are defined as acting on the elements; by Newton's third law, the forceson the nodes are equal and opposite, so a negative sign is needed. Viewing thesemidiscrete equations of motion in terms of Newton’s second law provides an intuitivefeel for these equations and is useful in remembering these equations.Initial Conditions.
Since the equations of motion are second order in time, initialconditions on the displacements and velocities are needed. The continuous form of theinitial conditions are given by Eqs. (2.2.22). In many cases, the initial conditions can beapplied by simply setting the nodal values of the variables to the initial values, i.e. bylettinguI (0 ) = u0 ( X I )∀Iand u˙I (0 ) = v0 ( X I )∀I(2.4.21)Thus the initial conditions on the nodal variables for a body which is initially at rest andundeformed areuI ( 0) = 0and u˙ I( 0) = 0 ∀ I(2.4.22)Least Square Fit to Initial Conditions. For more complex initial conditions, theinitial values of the nodal displacements and nodal velocities can be obtained by a leastsquare fit to the initial data. The least square fit for the initial displacements results fromminimizing the square of the difference between the finite element interpolate∑ N I ( X )uI (0) and the initial data u( X) .
LetM = 122Xb ∫Xa ∑ uI (0)NI ( X ) –u0 ( X ) ρ 0 A0 dX(2.4.23)IThe density is not necessary in this expression but as will be seen, it leads to equations interms of the mass matrix, which is quite convenient. To find the minimum setXb0 = ∂ M = ∫ NK ( X ) ∑ uI ( 0) N I( X )– u0( X) ρ0 A0 dX∂uK (0 ) X aI(2.4.24)Using the definition of the mass matrix, (2.4.11), it can be seen that the above can bewritten asMu(0 ) = g(2.4.25a)2-20T. Belytschko, Chapter 2, December 16, 1998XbgK = ∫X NK ( X )u0 ( X )ρ 0 A0 dX(2.4.25b)aThe least square fit to the initial velocity data is obtained similarly.
This method of fittingfinite element approximations to functions is often called an L 2 projection.Diagonal Mass Matrix. The mass matrix which results from a consistent derivationfrom the weak form is called a consistent mass matrix. In many applications, it isadvantageous to use a diagonal mass matrix called a lumped mass matrix. Procedures fordiagonalizing the mass matrix are often quite ad hoc, and there is little theory underlyingthese procedures. One of the most common procedures is the row-sum technique, inwhich the diagonal elements of the mass matrix are obtained byMIID =∑ MCIJ(2.4.26)Jwhere the sum is over the entire row of the matrix, MIJC is the consistent mass matrix andMIJD is the diagonal or lumped, mass matrix.The diagonal mass matrix can also be evaluated byMIID=∑Xb= ∫ ρ 0 NI ∑ N j A0dX = ∫ ρ0 N I A0 dX jXaXaXbCM IJJ(2.4.27)where we have used the fact that the sum of the shape functions must equal one; this is areproducing condition discussed in Chapter 8.
This diagonalization procedure conservesthe total momentum of a body, i.e. the momentum of the system with the diagonal mass isequivalent to that of the consistent mass, so∑ M IJCvJ =∑ M IIDvII, J(2.4.28)Ifor any nodal velocities.2.5 Relationships between Element and Global MatricesIn the previous section, we have developed the semidiscrete equations in terms ofglobal shape functions, which are defined over the entire domain, although they are usuallynonzero only in the elements adjacent to the node associated with the shape function. Theuse of global shape functions to derive the finite element equations provides littleunderstanding of how finite element programs are actually structured. In finite elementprograms, the nodal forces and the mass matrix are usually first computed on an elementlevel. The element nodal forces are combined into the global matrix by an operation calledscatter or vector assembly.
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