Javanbakht Zia - Computational Statics Revision Course

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A system of n linear independent equations for n unknowns can be generally written as The task is now to solve the above system to obtain the unknown . One of the classical approaches, the so-called Gaussian elimination, requires a gradual addition or subtraction of single equations in order to achieve a triangular structure at which the last equation contains only a single unknown. Back substitution allows to determine the remaining unknowns. Such a final structure may look as follows: In the scope of the finite element method, a matrix approach is more common for the solution of the system. The system () can be written in matrix form as: or in abbreviated form as: Multiplication from the left-hand side with the inverse of the coefficient matrix, i.e., , allows to solve the system: where is the identity matrix. It will be seen in the following chapters that the coefficient matrix corresponds to the so-called stiffness matrix , the column matrix to the matrix which contains the degrees of freedom ( ), and the right-hand side to the matrix which contains the nodal loads ( ). The application of Eq. () requires the calculation of the inverse of a matrix. This can be done for matrices of smaller dimensions as follows :

Equation for a matrix: Equation for a matrix: where Equation for a matrix: where and