Belytschko T. - Introduction (779635), страница 37
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Note the velocities are given by the same shape functionsince the shape functions are constant in time. The superposed dot on the nodal displacements isan ordinary derivative, since the nodal displacements are only functions of time.The accelerations are similarly given by the material time derivative of the velocities˙u˙ i ( X, t) = ˙u˙ iI ( t) N I (X )˙u˙ ( X,t ) = ˙u˙ I (t )N I ( X)or(4.4.6)It is emphasized that the shape functions are expressed in terms of the material coordinates in theupdated Lagrangian formulation even though we will use the weak form in the currentconfiguration.
As pointed out in Section 2.8, it is crucial to express the shape functions in terms ofmaterial coordinates when a Lagrangian mesh is used because we want the time dependence in thefinite element approximation of the motion to reside entirely in the nodal variables.The velocity gradient is obtained by substituting Eq. (4.4.5) into Eq.
(3.3.7), which yieldsLij = vi , j = viI∂N I= viI N I , j∂x jL = vINI, jor(4.4.7)and the rate-of-deformation is given byDij =12( Lij + Lji ) = 12 (viI N I, j + v jI N I ,i )(4.4.7b)In the construction of the finite element approximation to the motion, Eq. (4.4.1), we haveignored the velocity boundary conditions, i.e. the velocities given by Eq. (4.4.5) are not in thespace defined by Eq. (4.3.2). We will first develop the equations for an unconstrained body withno velocity boundary conditions, and then modify the discrete equations to account for the velocityboundary conditions.In Eq. (4.4.1), all components of the motion are approximated by the same shapefunctions.
This construction of the motion facilitates the representation of rigid body rotation,which is an essential requirement for convergence. This is discussed further in Chapter 8.The test function, or variation, is not a function of time, so we approximate the testfunction asδvi (X ) =δviI N I (X )orδv(X) = δv I NI (X)where δviI are the virtual nodal velocities.4-13(4.4.8)T. Belytschko, Lagrangian Meshes, December 16, 1998As a first step in the construction of the discrete finite element equations, the test function issubstituted into principle of virtual power giving∂N Iσ jidΩ− δviI ∫ N I ρbidΩ−∂xjΩΩδviI ∫nSD∑δviI ∫Γi=1tiN I tidΓ + δviI ∫ N I ρv˙ i dΩ= 0(4.4.9a)ΩThe stresses in (4.4.9a) are functions of the trial velocities and trial displacements.
From thedefinition of the test space, (4.3.4), the virtual velocities must vanish wherever the velocities areprescribed, i.e. δvi = 0 on Γvi and therefore only the virtual nodal velocities for nodes not on Γ viare arbitrary, as indicated above. Using the arbitrariness of the virtual nodal velocities everywhereexcept on Γ vi , it then follows that the weak form of the momentum equation isn SD∂NIN I ti dΓ + ∫ NI ρv˙ idΩ = 0∫Ω ∂x j σ ji dΩ− ∫Ω NI ρbidΩ− ∑∫Γt jj =1Ω∀I,i ∉Γvi(4.4.9b)However, the above form is difficult to remember. For purposes of convenience and for a betterphysical interpretation, it is worthwhile to ascribe physical names to each of the terms in the aboveequation.4.4.2.
Internal and External Nodal Forces. We define the nodal forces corresponding toeach term in the virtual power equation. This helps in remembering the equation and also providesa systematic procedure which is found in most finite element software. The internal nodal forcesare defined by∂ (δvi )∂Nσ ji dΩ= δviI ∫ I σ ji dΩ∂x j∂x jΩΩδ P int = δviI f iIint = ∫(4.4.10)where the third term is the definition of internal virtual power as given in Eqs. (B4.2.5) and(4.4.8) has been used in the last term.
From the above it can be seen that the internal nodal forcesare given by∂NIσ dΩ∂x j jiΩfiIint = ∫(4.4.11)These nodal forces are called internal because they represent the stresses in the body. Theseexpressions apply to both a complete mesh and to any element or group of elements, as has beendescribed in Chapter 2. Note that this expression involves derivatives of the shape functions withrespect to spatial coordinates and integration over the current configuration.
Equation (4.4.11) is akey equation in nonlinear finite element methods for updated Lagrangian meshes; it applies also toEulerian and ALE meshes.The external nodal forces are defined similarly in terms of the virtual external power4-14T. Belytschko, Lagrangian Meshes, December 16, 1998n SDδ P ext = δviI fiIext = ∫ δviρbidΩ + ∑ ∫ δvi ti dΓi =1 ΓtΩ= δviI ∫ NI ρbi dΩ+Ωi(4.4.12)n SD∑ δviI ∫ NI tidΓi =1Γtiso the external nodal forces are given byfiIext = ∫ N Iρbi dΩ+Ω∫ N t dΓI iorΓtif Iext = ∫ NI ρbdΩ+ ∫ N I e i ⋅ tdΓΩ(4.4.13)Γti4.4.3. Mass Matrix and Inertial Forces.
The inertial nodal forces are defined byδ P inert = δviI fiIinert = ∫ δviρv˙ i dΩ =δviI ∫ NI ρv˙ i dΩ(4.4.14)fiIinert = ∫ ρN I˙vi dΩ(4.4.15)ΩΩsoΩorf Iinert = ∫ ρN I ˙vdΩΩUsing the expression (4.4.6) for the accelerations in the above givesfiIinert = ∫ ρN I NJ dΩ ˙viJ(4.4.16)ΩIt is convenient to define these nodal forces as a product of a mass matrix and the nodalaccelerations. Defining the mass matrix byMijIJ = δij ∫ ρN I NJ dΩ(4.4.17)Ωit follows from (4.4.16) and (4.4.17) that the inertial forces are given byfiIinert = MijIJ˙v jJorf Iinert = M IJ ˙v J(4.4.18)4.4.4. Discrete Equations. With the definitions of the internal, external and inertial nodalforces, Eqs.
(4.4.10), (4.4.12) and (4.4.17), we can concisely write the discrete approximation tothe weak form (4.4.9a) asδviI ( fiIint − fiIext + MijIJv˙ jJ ) = 0 for ∀δviI ∉Γvi(4.4.19)Invoking the arbitrariness of the unconstrained, virtual nodal velocities givesMijIJv˙ jJ + f iIint = fiIext∀I,i ∉Γvior4-15M IJv˙ J + fIint = fIext(4.4.20)T. Belytschko, Lagrangian Meshes, December 16, 1998The above are the discrete momentum equations or the equations of motion; they are also calledthe semidiscrete momentum equations since they have not been discretized in time. The implicitsums are over all components and all nodes of the mesh; any prescribed velocity component thatappears in the above is not an unknown. The matrix form on the left depends on the interpretationof the indices: this is discussed further in Section 4.5.The semidiscrete momentum equations are a system of nDOF ordinary differential equationsin the nodal velocities, where nDOF is the number of nodal velocity components which areunconstrained; nDOF is often called the number of degrees of freedom.
To complete the system ofequations, we append the constitutive equations at the element quadrature points and the expressionfor the rate-of-deformation in terms of the nodal velocities. Let the nQ quadrature points in themesh be denoted by( )x Q( t) = N I XQ x I (t )(4.4.21)Note that the quadrature points are coincident with material points. Let nσ be the number ofindependent components of the stress tensor: in a two dimensional plane stress problem, nσ = 3 ,since the stress tensor σ is symmetric; in three-dimensional problems, nσ = 6 .The semidiscrete equations for the finite element approximation then consist of thefollowing ordinary differential equations in time:MijIJv˙ jJ + f iIint = fiIext for ( I,i ) ∉Γvi( ( ) )( )σ ij∇ X Q = Sij Dkl XQ , etc( ) (where Dij X Q = 21 Lij + L ji)(4.4.22)∀XQ(4.4.23)and Lij = N I , j ( XQ )viI(4.4.24)This is a standard initial value problem, consisting of first-order ordinary differential equations inthe velocities viI (t ) and the stresses σ ij ( XQ , t) .
If we substitute (4.4.24) into (4.4.23) to eliminatethe rate-of-deformation from the equations, the total number of unknowns is nDOF + nσ nQ . Thissystem of ordinary differential equations can be integrated in time by any of the methods forintegrating ordinary differential equations, such as Runge-Kutta methods or the central differencemethod; this is discussed in Chapter 6.The nodal velocities on prescribed velocity boundaries, viI , ( I,i ) ∈Γvi , are obtained fromthe boundary conditions, Eq. (B4.1.7b).
The initial conditions (B4.1.9) are applied at the nodesand quadrature pointsviI (0 ) = viI0()(4.4.25)( )σ ij XQ , 0 =σ ij0 XQ(4.4.26)4-16T. Belytschko, Lagrangian Meshes, December 16, 1998where viI0 and σ ij0 are initial data at the nodes and quadrature points. If data for the initialconditions are given at a different set of points, the values at the nodes and quadrature points canbe estimated by least square fits, as in Section 2.4.5.For an equilibrium problem, the accelerations vanish and the governing equations arefiIint = fiIext for ( I,i ) ∉Γvif int = f extor(4.4.27)along with (4.4.23) and (4.4.24). The above are called the discrete equilibrium equations.
If theconstitutive equations are rate-independent, then the discrete equilibrium equations are a set ofnonlinear algebraic equations in the stresses and nodal displacements. For rate-dependentmaterials, any rate terms must be discretized in time to obtain a set of nonlinear algebraic equations;this is further discussed in Chapter 6.4.4.5. Element Coordinates. Finite elements are usually developed with shape functionsexpressed in terms of parent element coordinates, which we will often call element coordinates forbrevity. Examples of element coordinates are triangular coordinates and isoparametric coordinates.We will next describe the use of shape functions expressed in terms of element coordinates.
Aspart of this description, we will show that the element coordinates can be considered an alternativeset of material coordinates in a Lagrangian mesh. Therefore, expressing the shape functions interms of element coordinates is intrinsically equivalent to expressing them in terms of materialcoordinates. We denote the parent element coordinates by ξie , or ξ e in tensor notation, and theparent domain by ∆ ; the superscript e will only be carried in the beginning of this description.The shape of the parent domain depends on the type of element and the dimension of the problem;it may be a biunit square, a triangle, or a cube, for example.
Specific parent domains are given inthe examples which follow.When a Lagrangian element is treated in terms of element coordinates, we are concernedwith three domains that correspond to an element:1. the parent element domain ∆ ;2. the current element domain Ω e = Ωe (t ) ;3. the initial (reference) element domain Ω e0The following maps are pertinent:1.
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