Consider the 3-D problem characterized by the functional II defined on the volume V 1 =...

Transcribed Image Text:

Consider the 3-D problem characterized by the functional II defined on the volume V 1 = Su = {( {( a ) + ( a ) + ( + u dv= (x,y,z)adv, Dy where a, , y and S are constant, u(x,y,z) is the unknown scalar field and (x,y,z) is the (known) "distributed load" function. a) Derive the finite element formulation (i.e., the local stiffness matrix and local load vector) for a M-node global element (i.e., in terms of the global coordinates x,y,z). As a preliminary step, describe the type of element needed for this problem (C, C, ...), the number of d.o.f. per node, the vector containing the nodal d.o.f. of the element and the expected size of the local stiffness matrix [k] and the local load vector {r}. [15 points] b) Give the expression of [k] and {r} for a 10-node isoparametric tetrahedral element, the parent element of which is defined by the local coordinates (, 1, () as in the figure (with 0 , n, 1). Write the expression of all the shape functions and derive the various quantities entering the expression of [k] 8 and {r} (you do not need to actually perform the 2 integrations!). [15 points] 4 1 9 10 6 7 3 n Consider the 3-D problem characterized by the functional II defined on the volume V 1 = Su = {( {( a ) + ( a ) + ( + u dv= (x,y,z)adv, Dy where a, , y and S are constant, u(x,y,z) is the unknown scalar field and (x,y,z) is the (known) "distributed load" function. a) Derive the finite element formulation (i.e., the local stiffness matrix and local load vector) for a M-node global element (i.e., in terms of the global coordinates x,y,z). As a preliminary step, describe the type of element needed for this problem (C, C, ...), the number of d.o.f. per node, the vector containing the nodal d.o.f. of the element and the expected size of the local stiffness matrix [k] and the local load vector {r}. [15 points] b) Give the expression of [k] and {r} for a 10-node isoparametric tetrahedral element, the parent element of which is defined by the local coordinates (, 1, () as in the figure (with 0 , n, 1). Write the expression of all the shape functions and derive the various quantities entering the expression of [k] 8 and {r} (you do not need to actually perform the 2 integrations!). [15 points] 4 1 9 10 6 7 3 n