# 数学代写|有限元方法代写Finite Element Method代考|ME672

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## 数学代写|有限元方法代写Finite Element Method代考|Assembly of Element Equations (Processor)

The assembly of element equations should be carried out as soon as the element matrices are computed, rather than waiting till element coefficients of all elements are computed. The latter requires storage of the element coefficients of all elements. In the former case, we can perform the assembly in the same loop in which a subroutine is called to calculate element matrices.
A feature of the finite element equations that enables us to save storage and computing time is the assembly of element matrices in upper-banded form. When element matrices are symmetric, the resulting global (or assembled) matrix is also symmetric, with many zeros away from the main diagonal.
Box 8.3.1: Fortran statements to compute element matrices.
$\begin{array}{ll}\text { C } \ \text { C } & \text { DO-LOOP on number of Gauss points begins here } \ \text { C }\end{array}$
DO $100 \mathrm{NI}=1$, NGP
$\mathrm{XI}=\operatorname{GAUSPT}(\mathrm{NI}, \mathrm{NGP})$
C
C Call subroutine SHAPE1D to evaluate interpolation functions
C and their global derivatives at the Gauss point XI
CALL SHAPE1D (H, IELEM, NPE, XI)
CONST $=$ GJ*GAUSWT $(N I, N G P)$
DO $30 \mathrm{~J}=1, \mathrm{NPE}$
30
$X=X+S F(J) \star \operatorname{ELX}(\mathrm{J})$
C
C Compute coefficient matrices for MODEL $=1$ and NTYPE $=0$
$\mathrm{CX}=\mathrm{CX} 0+\mathrm{CX}_1 \star \mathrm{X}$
$\mathrm{FX}=\mathrm{FX} 0+\mathrm{FX} 1 * \mathrm{X}+\mathrm{FX} 2 * \mathrm{X} * \mathrm{X}$
$\mathrm{AX}=\mathrm{AX} 0+\mathrm{AX} 1 * \mathrm{X}+\mathrm{AX} 2 * \mathrm{X} * \mathrm{X}+\mathrm{AX} 3 * \mathrm{X} * \mathrm{X} * \mathrm{X}$
DO $50 \mathrm{~J}=1, \mathrm{NPE}$
$\operatorname{ELF}(\mathrm{J})=\operatorname{ELF}(\mathrm{J})+\mathrm{CONST} * \mathrm{SF}(\mathrm{J}) * \mathrm{FX}$
DO $50 I=1, \mathrm{NPE}$
AIJ $=$ CONST*GDSF $(I) * G D S F(J)$
$\mathrm{CIJ}=\operatorname{CONST} * \mathrm{SF}(\mathrm{I}) * \mathrm{SF}(\mathrm{J})$
$\operatorname{ELK}(I, J)=\operatorname{ELK}(I, J)+A X \star A I J+C X * C I J$
50 100
CONTINUE

Therefore, it is sufficient to store only the upper half-band of the assembled matrix. The half bandwidth of a matrix is defined as follows. Let $N_i$ be the number of matrix elements between the diagonal element and the last nonzero element in the ith row, after which all elements in that row are zero; the half-bandwidth is the maximum of $\left(N_i+1\right)$
$$b_I=\max _{1 \leq i \leq n}\left(N_i+1\right)$$
where $n$ is the number of rows in the matrix (or equations in the problem). General-purpose equation solvers are available for such banded systems of equations.

## 数学代写|有限元方法代写Finite Element Method代考|Imposition of Boundary Conditions (Processor)

Imposition of boundary conditions on the primary and secondary global degrees of freedom can be carried out through a subroutine (BOUNDARY), which remains unchanged for two-dimensional or three-dimensional problems. There are three types of boundary conditions for any problem:

1. Essential boundary conditions, that is, boundary conditions on primary variables (Dirichlet boundary conditions).
2. Natural boundary conditions, that is, boundary conditions on secondary variables (Neumann boundary conditions).
3. Mixed (or Newton) boundary conditions (i.e., boundary conditions that relate the primary and secondary variables at a node).
The imposition of these three types of boundary conditions in a computational scheme is discussed next.

Specified primary variables
The procedure for implementing the boundary conditions on the primary variables involves modifying the assembled coefficient matrix (GLK) and right-hand-column vector (GLF) by three operations:

1. Moving the known products in each row to the right-hand side.
2. Replacing the columns and rows of GLK corresponding to the known primary variable by zeros, and setting the coefficient on the main diagonal to unity.
3. Replacing the corresponding component of the right-hand column by the specified value of the variable.

# 有限元方法代考

## 数学代写|有限元方法代写Finite Element Method代考|Assembly of Element Equations (Processor)

$\begin{array}{ll}\text { C } \ \text { C } & \text { DO-LOOP on number of Gauss points begins here } \ \text { C }\end{array}$

$\mathrm{XI}=\operatorname{GAUSPT}(\mathrm{NI}, \mathrm{NGP})$
c

CONST $=$ GJ*GAUSWT $(N I, N G P)$
DO $30 \mathrm{~J}=1, \mathrm{NPE}$
30.
$X=X+S F(J) \star \operatorname{ELX}(\mathrm{J})$
c
C计算MODEL $=1$和NTYPE $=0$的系数矩阵
$\mathrm{CX}=\mathrm{CX} 0+\mathrm{CX}_1 \star \mathrm{X}$
$\mathrm{FX}=\mathrm{FX} 0+\mathrm{FX} 1 * \mathrm{X}+\mathrm{FX} 2 * \mathrm{X} * \mathrm{X}$
$\mathrm{AX}=\mathrm{AX} 0+\mathrm{AX} 1 * \mathrm{X}+\mathrm{AX} 2 * \mathrm{X} * \mathrm{X}+\mathrm{AX} 3 * \mathrm{X} * \mathrm{X} * \mathrm{X}$
DO $50 \mathrm{~J}=1, \mathrm{NPE}$
$\operatorname{ELF}(\mathrm{J})=\operatorname{ELF}(\mathrm{J})+\mathrm{CONST} * \mathrm{SF}(\mathrm{J}) * \mathrm{FX}$
DO $50 I=1, \mathrm{NPE}$
AIJ $=$ CONST*GDSF $(I) * G D S F(J)$
$\mathrm{CIJ}=\operatorname{CONST} * \mathrm{SF}(\mathrm{I}) * \mathrm{SF}(\mathrm{J})$
$\operatorname{ELK}(I, J)=\operatorname{ELK}(I, J)+A X \star A I J+C X * C I J$
50 100

$$b_I=\max _{1 \leq i \leq n}\left(N_i+1\right)$$

## 有限元方法代写

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