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# 数学代写|有限元方法代写finite differences method代考|AMCS329

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## 数学代写|有限元代写Finite Element Method代考|Explicit and Implicit Formulations and Mass Lumping

The solution of the fully discretized equations of parabolic equations and hyperbolic equations (after assembly and imposition of boundary and initial conditions) require the inversion of $\hat{\mathbf{K}}$ appearing in Eqs. (7.4.29a) and (7.4.37a) to march forward in time and find the solution at different times. This can be an enormous computational expense, depending on the size of the mesh and the number of time steps. For example, if one needs to solve these equations for 1,000 time steps, the cost is equivalent to solving 1,000 static problems. Thus, it is of practical interest to find ways to reduce the computational cost. It is clear that if the element $\hat{\mathbf{K}}^e$ were a diagonal matrix, then the assembled or global coefficient matrix $\hat{\mathbf{K}}$ would be diagonal, and there is no inversion required to solve for $u_i^{s+1}$ (i.e., simply divide each equation with the diagonal element):
$$U_i^{s+1}=\frac{1}{\hat{K}{(i i)}}\left(\sum{j=1}^{N E Q} \bar{K}_{i j} U_j^s+\bar{F}_i^{s, s+1}\right) \quad(\text { no sum on } i)$$
Formulations that require the inversion of $\hat{\mathbf{K}}$ (because it is not diagonal) are termed implicit formulations and those in which no inversion is required are called explicit formulations.

In the finite element method, no time-approximation scheme results in a diagonal matrix $\hat{\mathbf{K}}$ because matrices $\mathbf{C}$ and/or $\mathbf{M}$ appearing in $\hat{\mathbf{K}}$ are not diagonal matrices. A matrix ( $\mathbf{C}$ or $\mathbf{M}$ ) computed according to the definition is called a consistent (mass) matrix, and it is not diagonal unless the approximation functions $\psi_i$ are orthogonal over the element domain. In realworld problems where hundreds of thousands of degrees of freedom are involved, the cost of computation precludes the inversion of large systems of equations. Thus, one needs to pick a scheme that eliminates $\mathbf{K}$ from $\hat{\mathbf{K}}$

(because, it would be a gross approximation to diagonalize $\mathbf{K}$ ) and then diagonalize $\mathbf{C}$ and/or $\mathbf{M}$ to have an explicit formulation.

For example, the forward difference scheme (i.e., $\alpha=0$ ) results in the following time-marching scheme [see Eq. (7.4.29a)]:
$$\mathbf{C U}^{s+1}=(\mathbf{C}-\Delta t \mathbf{K}) \mathbf{U}^s+\Delta t \mathbf{F}^s$$
If the matrix $\mathbf{C}$ is diagonal then the assembled equations can be solved directly (i.e., without inverting a matrix). Similarly, the central difference scheme for an undamped system (i.e., $\mathbf{C}=0$ ) is [see Eq. (7.4.42)]
$$\mathbf{M U}^{s+1}=(\Delta t)^2 \mathbf{F}^{s+1}+\left(2 \mathbf{M}-(\Delta t)^2 \mathbf{K}\right) \mathbf{U}^s-\mathbf{M} \dot{U}^s$$
which requires diagonalization of $\mathbf{M}$ (and $\mathbf{C}$, in the case of a damped system) in order for the central difference formulation to be explicit. The explicit nature of Eq. (7.4.48) motivated analysts to find rational ways of making $\mathbf{C}$ and/or $\mathbf{M}$ diagonal. There are several ways of constructing diagonal mass matrices by lumping the mass at the nodes, while preserving the total mass. Two such approaches are discussed next.

## 数学代写|有限元代写Finite Element Method代考|Row-sum lumping

The sum of the elements of each row of the consistent (mass) matrix is used as the diagonal element and setting the off-diagonal elements to zero [(ii) means no sum on $i]$ :
$$M_{(i i)}^e=\sum_{j=1}^n \int_{x_a^e}^{x_b^e} \rho \psi_i^e \psi_j^e d x=\int_{x_a^e}^{x_b^e} \rho \psi_i^e d x$$
where the property $\sum_{j=1}^n \psi_j^e=1$ of the interpolation functions is used.
When $\rho_e$ is element-wise constant, the consistent matrices associated with the linear and quadratic 1-D elements are
$$\mathbf{M}_{\mathrm{C}}^e=\frac{\rho_e A_e h_e}{6}\left[\begin{array}{ll} 2 & 1 \ 1 & 2 \end{array}\right], \quad \mathbf{M}_C^e=\frac{\rho_e A_e h_e}{30}\left[\begin{array}{rcr} 4 & 2 & -1 \ 2 & 16 & 2 \ -1 & 2 & 4 \end{array}\right]$$
As per Eq. (7.4.51), the associated diagonal matrices for the linear and quadratic elements are
$$\mathbf{M}_L^e=\frac{\rho_e A_e h_e}{2}\left[\begin{array}{ll} 1 & 0 \ 0 & 1 \end{array}\right], \quad \mathbf{M}_L^e=\frac{\rho_e A_e h_e}{6}\left[\begin{array}{lll} 1 & 0 & 0 \ 0 & 4 & 0 \ 0 & 0 & 1 \end{array}\right]$$
Here subscripts $L$ and $C$ refer to lumped and consistent mass matrices, respectively.
The consistent mass matrix for the Euler-Bernoulli beam is given in Eq. (7.3.57). The row-sum diagonal mass matrix is obtained in two ways: (a) neglecting the terms corresponding to the rotational degrees of freedom in each row and (b) neglecting the terms corresponding to the rotational degrees of freedom in rows 1 and 3 and neglecting the terms associated with the translational degrees of freedom in rows in 2 and 4 :
$$\mathbf{M}_L^e=\frac{\rho_e A_e h_e}{2}\left[\begin{array}{llll} 1 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 0 \end{array}\right], \quad \hat{\mathbf{M}}_L^e=\frac{\rho_e A_e h_e}{420}\left[\begin{array}{rrrr} 210 & 0 & 0 & 0 \ 0 & h_e^2 & 0 & 0 \ 0 & 0 & 210 & 0 \ 0 & 0 & 0 & h_e^2 \end{array}\right]$$
The consistent mass matrix of the Timoshenko beam theory is given in Eq. (7.3.63b). The lumped mass matrices for the Timoshenko beam are
$$\mathbf{M}_L^e=\frac{\rho_e A_e h_e}{2}\left[\begin{array}{llll} 1 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 0 \end{array}\right], \quad \hat{\mathbf{M}}_L^e=\frac{\rho_e A_e h_e}{2}\left[\begin{array}{cccc} 1 & 0 & 0 & 0 \ 0 & r_e & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & r_e \end{array}\right], \quad r_e=\frac{I_e}{A_e}$$

## 数学代写|有限元代写Finite Element Method代考|Explicit and Implicit Formulations and Mass Lumping

$$U_i^{s+1}=\frac{1}{\hat{K}{(i i)}}\left(\sum{j=1}^{N E Q} \bar{K}_{i j} U_j^s+\bar{F}_i^{s, s+1}\right) \quad(\text { no sum on } i)$$

(因为，这将是对角化$\mathbf{K}$的粗略近似值)，然后对角化$\mathbf{C}$和/或$\mathbf{M}$以获得显式公式。

$$\mathbf{C U}^{s+1}=(\mathbf{C}-\Delta t \mathbf{K}) \mathbf{U}^s+\Delta t \mathbf{F}^s$$

$$\mathbf{M U}^{s+1}=(\Delta t)^2 \mathbf{F}^{s+1}+\left(2 \mathbf{M}-(\Delta t)^2 \mathbf{K}\right) \mathbf{U}^s-\mathbf{M} \dot{U}^s$$

## 数学代写|有限元代写Finite Element Method代考|Row-sum lumping

$$M_{(i i)}^e=\sum_{j=1}^n \int_{x_a^e}^{x_b^e} \rho \psi_i^e \psi_j^e d x=\int_{x_a^e}^{x_b^e} \rho \psi_i^e d x$$

$$\mathbf{M}_{\mathrm{C}}^e=\frac{\rho_e A_e h_e}{6}\left[\begin{array}{ll} 2 & 1 \ 1 & 2 \end{array}\right], \quad \mathbf{M}_C^e=\frac{\rho_e A_e h_e}{30}\left[\begin{array}{rcr} 4 & 2 & -1 \ 2 & 16 & 2 \ -1 & 2 & 4 \end{array}\right]$$

$$\mathbf{M}_L^e=\frac{\rho_e A_e h_e}{2}\left[\begin{array}{ll} 1 & 0 \ 0 & 1 \end{array}\right], \quad \mathbf{M}_L^e=\frac{\rho_e A_e h_e}{6}\left[\begin{array}{lll} 1 & 0 & 0 \ 0 & 4 & 0 \ 0 & 0 & 1 \end{array}\right]$$

$$\mathbf{M}_L^e=\frac{\rho_e A_e h_e}{2}\left[\begin{array}{llll} 1 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 0 \end{array}\right], \quad \hat{\mathbf{M}}_L^e=\frac{\rho_e A_e h_e}{420}\left[\begin{array}{rrrr} 210 & 0 & 0 & 0 \ 0 & h_e^2 & 0 & 0 \ 0 & 0 & 210 & 0 \ 0 & 0 & 0 & h_e^2 \end{array}\right]$$
Timoshenko梁理论的一致质量矩阵如式(7.3.63b)所示。Timoshenko梁的集总质量矩阵为
$$\mathbf{M}_L^e=\frac{\rho_e A_e h_e}{2}\left[\begin{array}{llll} 1 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 0 \end{array}\right], \quad \hat{\mathbf{M}}_L^e=\frac{\rho_e A_e h_e}{2}\left[\begin{array}{cccc} 1 & 0 & 0 & 0 \ 0 & r_e & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & r_e \end{array}\right], \quad r_e=\frac{I_e}{A_e}$$

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