Global stiffness matrix

Therefore the assembled global stiffness matrix in this case is written as... 

C ASSEMBLE GLOBAL STIFFNESS MATRIX IN A BANDED FORM C... 

The modeling of complex solids has greatly advanced since the advent, around 1960, of the finite element method [196], Here the material is divided into a number of subdomains, termed elements, with associated nodes. The elements are considered to consist of materials, the constitutive equations of which are well known, and, upon change of the system, the nodes suffer nodal displacements and concomitant generalized nodal forces. The method involves construction of a global stiffness matrix that comprises the contributions from all elements, the relevant boundary conditions and body and thermal forces a typical problem is then to compute the nodal displacements (i. e., the local strains) by solving the system K u = F, where K is the stiffness matrix, u the... 

The next step in the numerical implementation is to find a relation between each individual element stiffness matrix, element displacement vector and element force vector and the global system. Hence, we must generate a global stiffness matrix, global displacement vector and global force vector. This step in the implementation is actually only data management or book keeping. 

To use computer storage more efficiently, the vector of unknown temperatures will eventually be stored in the global force vector, f. The next steps in the finite element procedure (Table 9.1) will be to form the global stiffness matrix and force vector, and to solve the resulting linear system of algebraic equations, as presented in Algorithm 5. 

The global stiffness matrix and force vector, which represent our equation system are formed by direct addition of the element stiffness matrices and force vectors. The corresponding position of an element component in the global system is given by the connectivity matrix. In two- or three-dimensional problems the positions are related to the connectivity matrix as well as the direction under consideration. The global stiffness matrix and force vector assembly technique is presented in Algorithm 6. 

Although this algorithm is clear and simple, it presents the most ineffective way of storing the global stiffness matrix since it results in a full sparse matrix. Later in this section we will discuss how the storage space and computation time is minimized by using alternative storing schemes such as banded matrices. 

Algorithm 6 Global stiffness matrix and force vector assembly... 

Before we proceed to our discussion of global stiffness matrix storage schemes, we will discuss the last aspect of the finite element implementation, namely, the application of the boundary conditions. As discussed earlier, the natural boundary conditions are imbedded in the finite element equation system - it is implied that every boundary node without an... 

Algorithm 8 Compute the bandwidth of a banded Global Stiffness matrix... 

The global "force" vector F is evaluated from the boundary conditions. The skyline solver (2) is adopted to solve equation (12). The main purpose of skyline solver is to find the LU decomposition of the global "stiffness" matrix while using minimum storage space. We concentrate meshes near z =0 in z direction but spread evenly in x direction because dislocations are spaced evenly in x direction along z =0. 

Add element stiffness matrices into global stiffness matrix... 

Assemble the tool removal global stiffness matrix, [K r], and set the tool removal global force vector, Ftr), to zero. 

The Newton-Raphson method may be computationally expensive in a multi dof problem because a new global stiffness matrix is used in each iterative step. In the Modified Newton-Raphson method, the same global stiffness matrix is used in all the iterative steps within an increment. This method requires more iterations to achieve convergence but each iteration is computed far more quickly. 

The shown tapered bar is discretized by four linear elements of linearly changing cross sections. The bar is fixed at the left side and the right end is subjected to a force boundary condition F. The two left and right elements, i.e., elements (I + II) and elements (III + IV), should be grouped to substructures and used to solve the problem. The classical solution procedure without substructures would consist of assembling the global stiffness matrix of all four elements which would result in the following 5x5 stiffness matrix ... 

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