Discussion topic for Module 1 of the course Introduction to Finite Element Analysis In the notes for module 1, I’ve argued that the global equation set, KD = F, is singular, and thus not solvable, as is. Singular, to review, means that the matrix K has no inverse, hence we cannot isolate D on the left hand side for solution. I went on to state that we rectify the situation by asserting boundary conditions, after which we have a set of equations KD = F, which IS solvable. So, it seems obvious to me that imposition of boundary conditions somehow alters the system of equations on some fundamental level. My question to you for discussion is this: how does the imposition of boundary conditions make solution possible, and what would be the implications of asserting different set of boundary conditions? Please back this up with a conceptual example, where boundary conditions have a practical engineering interpretation. As a concrete example, consider the following set of equations: 26 4 3 9 2 3 1 7 4 3 1 4 3 2 1 3 21 4 2 3 4 2 3 1 7 x x x x x x x x x x x x x which can be written in matrix-vector form as 26 7 1 21 4 3 2 1 1 9 3 1 1 1 0 0 1 2 0 3 2 4 3 7 x x x x The equation set above is NOT a result of any finite element analysis, it was merely chosen by me to make a point. You may note, however, that the fourth line of the matrix is a combination of the first three (specifically, it’s the first line, minus twice the second line, plus the third line.) As a result, the set has a zero determinant, and thus the inverse of the matrix does not exist. Now, I submit that if we assert one value for any of the “x” terms, we come to a solvable set. For example, if you try 1 1 x , you can then solve to get 2 2 x , 3 3 x and 4 4 x . But what if you instead assert x 2 = 1? How does this relate to practical engineering situations? I would like to see one answer to this question from each group, preferably before May 30 Please delegate one group member to send me a response, and I will look it over and get back to you. I have arranged to have my response to this question posted on May 30. If you wish to have more personal feedback from me, please contact me, or send your group response.

EL507 Intro to Finite Element Analysis

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Discussion topic for Module 1 of the course Introduction to Finite Element Analysis

In the notes for module 1, I’ve argued that the global equation set, KD = F, is singular, and thus not solvable, as is. Singular, to review, means that the matrix K has no inverse, hence we cannot isolate D on the left hand side for solution. I went on to state that we rectify the situation by asserting boundary conditions, after which we have a set of equations KD = F, which IS solvable. So, it seems obvious to me that imposition of boundary conditions somehow alters the system of equations on some fundamental level. My question to you for discussion is this: how does the imposition of boundary conditions make solution possible, and what would be the implications of asserting different set of boundary conditions? Please back this up with a conceptual example, where boundary conditions have a practical engineering interpretation. As a concrete example, consider the following set of equations:

26439231

743

143213

2142342317

xxxx

xxx

xxxx

which can be written in matrix-vector form as

1931

1100

1203

2437

The equation set above is NOT a result of any finite element analysis, it was merely chosen by me to make a point. You may note, however, that the fourth line of the matrix is a combination of the first three (specifically, it’s the first line, minus twice the second line, plus the third line.) As a result, the set has a zero determinant, and thus the inverse of the matrix does not exist. Now, I submit that if we assert one value for any of the “x” terms, we come to a solvable set. For example, if you try 11 x , you can then solve to get 22 x , 33 x and 44 x .

But what if you instead assert x2 = 1? How does this relate to practical engineering situations? I would like to see one answer to this question from each group, preferably before May 30 Please delegate one group member to send me a response, and I will look it over and get back to you. I have arranged to have my response to this question posted on May 30. If you wish to have more personal feedback from me, please contact me, or send your group response.