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数学代写|有限元方法代写finite differences method代考|Assembly of Element Equations

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数学代写|有限元代写Finite Element Method代考|Assembly of Element Equations

In deriving the element equations, we isolated a typical element from the mesh and formulated the weak form and developed its finite element model. The finite element model of a typical element contains $n$ equations among $n+$ 2 unknowns, $\left(u_1^e, u_2^e, \ldots, u_n^e\right)$ and $\left(Q_1^e, Q_n^e\right)$. Hence, they cannot be solved without using the equations from other elements. From a physical point of view, this makes sense because one should not be able to solve the element equations without considering the assembled set of equations and the boundary conditions of the total problem.
To obtain the finite element equations of the total problem, we must put the elements back into their original positions. In putting the elements with their nodal degrees of freedom back into their original positions, we must require that the primary variable $u(x)$ is uniquely defined (i.e., $u$ is continuous) and the source terms $Q_i^e$ are “balanced” at the points where elements are connected to each other. Of course, if the variable $u$ is not continuous, we do not impose its continuity; but in all problems studied in this book, unless otherwise stated explicitly (like in the case of an internal hinge in the case of beam bending), the primary variables are required to be continuous. Thus, the assembly of elements is carried out by imposing the following two conditions:

If the end node $i$ of element $\Omega^e$ is connected to the end node $j$ of element $\Omega^f$ and the end node $k$ of element $\Omega^g$, the continuity of the primary variable $u$ requires
$$u_i^{(e)}=u_j^{(f)}=u_k^{(g)}$$
When node $n$ of element $\Omega e$ is connected to node 1 of element $\Omega^{e+1}$ (with $m$ nodes) in series, as shown in Fig. 3.4.10a, the continuity of $u$ requires
$$u_n^{(e)}=u_1^{(e+1)}$$

For the same three elements, the balance of secondary variables at connecting nodes requires
$$Q_i^{(e)}+Q_j^{(f)}+Q_k^{(g)}=Q_I$$
where $I$ is the global node number assigned to the nodal point that is common to the three elements, and $Q_I$ is the value of externally applied source, if any (otherwise zero), at this node (the sign of $Q_I$ must be consistent with the sign of $Q_e$ in Fig. 3.4.4). For the case shown in Fig. 3.4.10, we have
$$Q_n^e+Q_1^{e+1}= \begin{cases}0, & \text { if no external point source is applied } \ Q_l, & \text { if an external point source of magnitude } \ Q_l \text { is applied }\end{cases}$$

数学代写|有限元代写Finite Element Method代考|Postprocessing of the Solution

The solution of the finite element equations in Eq. (3.4.59) gives the values of the primary variables (e.g., displacement, velocity, or temperature) at the global nodes. Once the nodal values of the primary variables are known, we can use the finite element approximation $u_h^e(x)$ to compute the desired quantities. The process of computing desired quantities in numerical form or graphical form from the known finite element solution is termed postprocessing; this phrase is meant to indicate that further computations are made after obtaining the solution of the finite element equations for the nodal values of the primary variables.
Postprocessing of the solution includes one or more of the following tasks:

1. Computation of the primary and secondary variables at points of interest; primary variables are known at nodal points.
2. Interpretation of the results to check whether the solution makes sense (an understanding of the physical process and experience are the guides when other solutions are not available for comparison).
3. Tabular and/or graphical presentation of the results.
To determine the solution $u$ as a continuous function of position $x$, we return to the approximation in Eq. (3.4.28) over each element:
$$u(x) \approx\left{\begin{array}{l} u_h^1(x)=\sum_{j=1}^n u_j^1 \psi_j^1(x) \ u_h^2(x)=\sum_{j=1}^n u_j^2 \psi_j^2(x) \ \vdots \ u_h^N(x)=\sum_{j=1}^n u_j^N \psi_j^N(x) \end{array}\right.$$

数学代写|有限元代写Finite Element Method代考|Assembly of Element Equations

$$u_i^{(e)}=u_j^{(f)}=u_k^{(g)}$$

$$u_n^{(e)}=u_1^{(e+1)}$$

$$Q_i^{(e)}+Q_j^{(f)}+Q_k^{(g)}=Q_I$$

$$Q_n^e+Q_1^{e+1}= \begin{cases}0, & \text { if no external point source is applied } \ Q_l, & \text { if an external point source of magnitude } \ Q_l \text { is applied }\end{cases}$$

数学代写|有限元代写Finite Element Method代考|Postprocessing of the Solution

$$u(x) \approx\left{\begin{array}{l} u_h^1(x)=\sum_{j=1}^n u_j^1 \psi_j^1(x) \ u_h^2(x)=\sum_{j=1}^n u_j^2 \psi_j^2(x) \ \vdots \ u_h^N(x)=\sum_{j=1}^n u_j^N \psi_j^N(x) \end{array}\right.$$

MATLAB代写

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