Mechanical Engineering

Advanced Applied Finite Element Methods by Carl T. F. Ross

By Carl T. F. Ross

This publication is geared toward senior undergraduates, graduates and engineers. It fills the distance among the various textbooks on conventional utilized Mechanics and postgraduate books on Finite aspect equipment.

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48 Matrix algebra [Ch. 73). ^22*2 + /*23*3 + . . + A\„X„ = C2l A\2 x2 + A j-, x, + . . + A I, xn = C\ '. 73) An2X2 + ^ n 3 * 3 + · · · · + AnnX„ = C„ The above process can be repeated by eliminating xx, x3, etc. nxn = + B„nx„ = kn . 74) BLxn = kxn . 74), etc. the other unknowns can be determined. 71). 76). 76). 76). If this process is repeated to the nth equation, then the following set of simultaneous equations is obtained: Sec. 77). Multiply this equation by A\2, A\2, etc. and take these equations away from the third, fourth, etc.

87) Solution of the simultaneous equations is carried out by the triangulation process described in Sec. 3, except that the property of symmetry is used to save space. 87). JJ>/Rai,l> JK=0 FOR KK=J,T TO NW JK=JK+1 AC IK, JK>=fiaK, JK>-CN*fii 11, KK> NEXT KK A a I.. T=JJ+1 5ΘΘ 510 520 530 540 550 560 570 c a i :>=ca1 :>-R< Ι Ι ,KK>*CCJJ> NEXT KK GOTO 45Θ PRINT"THE VECTOR ON THE LHS C O FOR 11=1 TO N PRINT C a p NEXT II END IS-" 54 Matrix algebra [Ch. 1 The advantages of this method of solution can readily be seen, both from the point of view of saving space and time, and also because of the increased precision obtained.

1 Method of successive transformation (Gauss-Jordan) This is a pivotal method and is somewhat similar to Chio's method for evaluating determinants. Simultaneous equations are sometimes solved by using the property of an inverse matrix, and this principle is used to formulate the process for the present method. 32) •^nl·*! 32) can be rewritten as follows: [A){x} = [C]. 33) by [/I] - 1 , the following is obtained: [A]-l[A]{x} = [A)-HQ. 34) becomes {x} = [A)-i{Q . 37) therefore {x} = [B){C} . 35) shows that [B] = \A\~l .

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