9 Parameter Triangular Plate Element Study

The convergence criteria of the finite element require its elementfunCtion satisfy the completeness and continuity For numerous Cl continuityelement, it is difficult to satisfy the requirement. In order to loose therequirement of continuity as possible, study the construCtion mechanism andpropedy of Cl cofltinuity element, enlnye the construCting area of finiteelement, explore simpler, higher efficient element, based on the weak form ofthe domain equilibrium in this paper, obtains integral continuity condition,weak continuity condition gradually enlarge the construCting area of finiteelement indeed, concludes that their average stralns are identical.Because it is difficult to study non-conforming element by the principleof minimum potential energy, this paper re-deduce the basic equations ofelasticity and thin plate bending by the opinion of domain equilibrium, and itsweighted form is just the weak form of corresponding differential equation, byit could deduce corresponding principle of energy too. But it meansequilibrium and continuity in an area, not point by point all over the are4hence could loose the continuity requirement conveniently.The weighted integral continuity and integral continuity conditions canbe obtained by the weak form of the differential equation. Because the finiteelement in integral continuity converges, it explains the coniinuity pointwisealong the inter-element boundary too exaCting and unnecessary. The finiteelements satisfied the integral condition are quasi-conforming element,generalized conforming element, and elements constructed by 2 set parametermCthod, and they are equivalent in some sense, and all converge to constanistrain. The constant strain and node error could be obtained easily withconstruCting the element function of the quasi-conforming element. Loosedthe illter-elemefit colltinuity further with the requirement of Irons' patch teSt,obtains the weak continuity condition ensured convergence. It comprises thefunCtion value at the node continue and its nomina1 derivative continue 2conditions, hence do enlarge the construCting area of the finite element.Modify the non-conformin8 nominal derivation of BCIZ element accordin8 tothe weak continuity conditiory obtained convergent element too.Examined the finite element in weak continuity, they possess identicalaverage stain. Other convergent finite elemeot, such as discfate kirchhoff plateelement (DKT), POssess' same conStallt Strain too. If the element strain can beexPressed as linear polynomial, the element converge to its conStant sttaln isproved in the paPer The BCIZ element is juSt satisfied the constant sttalncondition in partial regUlar mesh too. The average strain in weak continuitycan become a method to modify non-conver8ent non-confOrming element. Forexample, modify BCIZ elemeat's un-correCt cond stfain obtain convergentelement too. For the element function with quartic shaPe function in weakcontinuity, its average straln is equals to the constan Stain with cubic shaPefunctiory and can obtained convergent element stiffhess matrix by 3 pointshammer integration, approach to the quadratic Sttaln as linear Strain can provethis. Because of the characteristic that all the convergefit element function'saverages strain are identical, could used it as a method to test whether anelement is convergent. For example, could explain the phenomena that theBCIZ element is convergent in part roplar meshes, and could modifyno-convergeot element by average strain.To construCt high precision, simple, high comPutational efficient element,two complete cubic shaPe funCtion elements were construCted is the paperThe one was a geometrical symmetry element in orthogonal coordinate, theOther was a geomstrical asymmetry element in area coordinate. Both possessgood computational precision, but not in coarse mesh. To look for bcherelement, numerical comPutations for l9 kind of 9 parameter triangle p...