Belytschko T. - Introduction (779635), страница 11

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The mass matrix and other square matrices are combinedfrom the element level to the global level by an operation called matrix assembly. Whenthe nodal displacements are needed for computations, they are extracted from the global2-21T. Belytschko, Chapter 2, December 16, 1998matrix by an operation called gather. These operations are described in the following.

Inaddition we will show that there is no need to distinguish element and global shapefunctions and element and global equations for the nodal forces: the expressions areidentical and the element related expressions can always be obtained by limiting theintegration to the domain of the element.The relations between element matrices and the corresponding global matrices willobtained by the use of the connectivity matrices Le.

The nodal displacements and nodalforces of element e are denoted by u e and fe , respectively, and are column matrices oforder m, where m is the number of nodes per element. Thus for a 2-node element, theelement nodal displacement matrix is uTe = [u1 , u2 ]e . The corresponding element nodalforce matrix is f Te = [ f 1 , f 2 ]e . We will place the element identifier “e” as either asubscript or superscript, but will always use the letter “e” for the purpose of identifyingelement-related quantities.The element and global nodal forces must be defined so that their scalar products withthe corresponding nodal displacement increments gives an increment of work.

This wasused in defining the nodal forces in Section 2.4. In most cases, meeting this requiremententails little beyond being careful to arrange the nodal displacements and nodal forces in thesame order in the corresponding matrices. This feature of the nodal force and displacementmatrices is crucial to the assembly procedure and symmetry of linear and linearizedequations.The element nodal displacements are related to the global nodal displacements byδue = Leδuue = Le u(2.5.1)The matrix Le is a Boolean matrix, i.e. it consists of the integers 0 and 1.

An example ofthe Le matrix for a specific mesh is given later in this Section. The operation of extractingue from u is called a gather because in this operation the small element vectors aregathered from the global vector.The element nodal forces are defined analogously to (2.4.4) as those forces which givethe internal work:δWinte= δuTe f inte=∫X emX1eδu, X PA0 dX(2.5.2)To obtain the relations between global and local nodal forces, we use the fact that the totalvirtual internal energy is the sum of the element internal energies:δWint =∑ δWinteorδuT f int =e∑ δu fT inte e(2.5.3)eSubstituting (2.5.1) into the (2.5.3) yieldsδuT f int = δuT∑L fT inte e(2.5.4)e2-22T. Belytschko, Chapter 2, December 16, 1998Since the above must hold for arbitrary δu, it follows thatf int =∑L fT inte e(2.5.5)ewhich is the relationship between element nodal forces and global nodal forces.

The aboveoperation is called a scatter, for the small element vector is scattered into the global arrayaccording to the node numbers. Similar expressions can be derived for the external nodalforces and the inertial forcesf ext =∑L fT exte e ,ef inert =∑ LTefeinert(2.5.6)eThe gather and scatter operations are illustrated in Fig. 2 for a one dimensional meshof two-node elements. The sequence of gather, compute and scatter is illustrated for twoelements in the mesh.

As can be seen, the displacements are gathered according to the nodenumbers of the element. Other nodal variables, such as nodal velocities and temperatures,can be gathered similarly. In the scatter, the nodal forces are then returned to the globalforce matrix according to the node numbers. The scatter operation is identical for the othernodal forces.2-23T.

Belytschko, Chapter 2, December 16, 1998456u1u2e1e1f1f217 [ 2 ]1314151617181920u1u2e1e2e23 [ 2]f1f2e2[ ] = local node numbersGATHERCOMPUTEu e = L euf45678910111214 [ 1]7891011121235 [ 1]123SCATTER= Se LeT fe1314151617181920fuFig.

2.2. Illustration of gather and scatter for a one-dimensional mesh of two-node elements, showing thegather of two sets of element nodal displacements and the scatter of the computed nodal forces.In order to describe the assembly of the global mass matrix from the element massmatrices, the element inertial nodal forces are defined as a product of an element massmatrix and the element acceleration, similarly to (2.4.13):feinert = Me ae(2.5.7)By taking time derivatives of Eq. (2.5.1), we can relate the element and global accelerationsby ae = Le a,(the connectivity matrix does not change with time) and inserting this into theabove and using (2.5.6) yieldsf inert =∑ LTeMe Le a(2.5.8)eComparing (2.5.8) to (2.4.13), it can be seen that the global mass matrix is given in termsof the element matrices by2-24T.

Belytschko, Chapter 2, December 16, 1998M=∑L M LTee(2.5.9)eeThe above operation is the well known procedure of matrix assembly. This is the sameoperation which is used to assemble the stiffness matrix from element stiffnesses in linearfinite element methods.N2N112e3eN1N212Fig. 2.3. Illustration of element N e (X) and global shape functions N(X) for a one dimensional mesh oflinear displacement, two-node elements.Relations between element shape functions and global shape functions can also bedeveloped by using the connectivity matrices.

However, we shall shortly see that in mostcases there is no need to distinguish them. The element shape functions are defined as theinterpolants Ne ( X ) , which when multiplied by the element nodal displacements, give thedisplacement field in the element, i.e. the displacement field in element e is given bymu ( X ) = N ( X )ue = ∑ N eI ( X )u eIee(2.5.10)I =1The global displacement field is obtained by summing the displacement fields for allelements, which givesnene m nNu( X ) = ∑ Ne ( X )L eu = ∑ ∑ ∑ N eI ( X )LeIJu J(2.5.11)e =1 I =1 J =1e= 1where Eq.

(2.5.1) has been used in the above. Comparing the above with Eq. (2.4.1), wesee thatneN( X ) = ∑ N ( X)L e or N J ( X) =e= 1ene m∑ ∑ NIe( X) LeIJe =1 I =12-25(2.5.12)T. Belytschko, Chapter 2, December 16, 1998Thus the global shape functions are obtained from the element shape functions bysumming according to the node numbers of the elements.

This relationship is illustratedgraphically for a two-node linear displacement element in Fig. 2.3.We will now show that the expressions for the element nodal forces are equivalent tothe global nodal forces, except that the integrals are restricted to the elements. Using Eq.(2.5.2) and the element form of the displacement field, we obtainδWein t= δu Te feinteT Xm= δue eX1∫N e, X PA0dX(2.5.13)Invoking the arbitrariness of the virtual nodal displacements, we obtainfeint = ∫XemX1eN , X PA0 dX or fIint,e = ∫XmeX1eNI , X PA0 dX(2.5.14)where the superscript e has been removed from the last expression since in element e,Ne ( X ) = N( X) .Comparing the above with (2.4.6), we can see that (2.5.14) is identical to the globalexpression (2.4.6) except that integrals here are limited to an element.

Identical results canbe obtained for the mass matrix and the external force matrix. Therefore, in subsequentderivations we will usually not distinguish element and global forms of the matrices: theelement forms are identical to the global forms except that element matrices correspond tointegrals over the element domain, whereas global force matrices correspond to integralsover the entire domain.In finite element programs, global nodal forces are not computed directly but obtainedfrom element nodal forces by assembly, i.e.

the scatter operation. Furthermore, theessential boundary conditions need not be considered until the final steps of the procedure.Therefore we will usually concern ourselves only with obtaining the element equations.The assembly of the element equations for the complete model and the imposition ofboundary conditions is a standard procedure.We will often write the internal nodal force expressions for the total Lagrangianformulation in terms of a B0 matrix, where B0 is in the one-dimensional case a row matrixdefined byB0 I = NI , X(2.5.15)The nought is specifically included to indicate that the derivatives are with respect to theinitial, or material, coordinates.

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