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Tham khảo tài liệu 'nonlinear finite elements for continua and structures part 11', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | W.K.Liu Chapter 7 26 formulations are identical to those derived in Chapter 4. However note that the spatial domain now depends on how the mesh motion is updated which is one of the key ingredients in the updated ALE formulation. The variational equations corresponding to the conservation equations of Box 7.1 are obtained by multiplying by the test functions Sp and SVị integrating over the spatial domain Q and employing the divergence theorem to embed the traction force vector t on the boundary rt. Following the same procedures as in Chapter 4 we can achieve the following weak forms Continuity Equation JQspp yQ jQ Spp t X dQ jQsp cpd -jQsppVMdQ 7.16.2a Momentum Equation jQS rpEdQ jQSvipV- t XdQ jQSv pcv dQ 7.16.2b -JQS V-j ơidQ JQS VpbdQ x Jr tS Vd It is noted that because convective terms p ịCị and Vị jCj appeared in the continuity and momentum equations a Galerkin finite element formulation will give rise to numerical difficulties. Therefore in this section the Petrov-Galerkin formulation will be employed to alleviate some of these difficulties. In a Petrov Galerkin finite element discretization the current domain Q is subdivided into elements. However different sets of shape functions N and Np for the trial functions and N and N p for the test functions will be used to interpolate the velocity and density respectively. If N N the Galerkin ALE formulations will be obtained. The choice of N and N p to eliminate numerical oscillations will be described in section . The finite element matrix equations corresponding to Eq. 7.16.2a b are Continuity equation M pp t x Lpp Kpp 0 7.16.3a where Mp Lp Kp are generalized mass convective and stiffness matrices respectively for density under a reference description such that Mp MJ J NfNjdQ 7.16.3b JQ Lp i j JQ Nt cN dQ 7.16.3c W.K.Liu Chapter 7 27 Kp Kfj ju Nf ViN dU 7.16.3d Momentum Equation Mv t x Lv fint fext 7.16.4a where M and L are generalized mass and convective matrices respectively for velocity under a reference .
