SMS – Implicit Euler

Differential equation

\displaystyle \ddot{\mathbf{x}} = \mathbf{f}(\mathbf{x}, \dot{\mathbf{x}})

Reformulated differential equation (now 1st order)

\displaystyle \left(\begin{array}{c}\dot{\mathbf{x}}\\ \dot{\mathbf{v}} \end{array}\right) = \left(\begin{array}{c}\mathbf{v}\\ \mathbf{f}(\mathbf{x}, \mathbf{v}) \end{array}\right)

Explicit Euler

\displaystyle \left(\begin{array}{c}\Delta\mathbf{x}\\ \Delta\mathbf{v} \end{array}\right) = h\left(\begin{array}{c}\mathbf{v}_0\\ \mathbf{f}(\mathbf{x}_0, \mathbf{v}_0) \end{array}\right) = h\left(\begin{array}{c}\mathbf{v}_0\\ \mathbf{f}_0 \end{array}\right)

Implicit Euler

\displaystyle \left(\begin{array}{c}\Delta\mathbf{x}\\ \Delta\mathbf{v} \end{array}\right) = h\left(\begin{array}{c}\mathbf{v}_0 + \Delta\mathbf{v}\\ \mathbf{f}(\mathbf{x}_0 + \Delta\mathbf{x}, \mathbf{v}_0 + \Delta\mathbf{v}) \end{array}\right)

The only problematic equation

\displaystyle \Delta\mathbf{v} = h\,\mathbf{f}(\mathbf{x}_0 + h(\mathbf{v}_0 + \Delta\mathbf{v}), \mathbf{v}_0 + \Delta\mathbf{v})

First order expansion of \mathbf{f} around (\mathbf{x}_0, \mathbf{v}_0)

\displaystyle \mathbf{f}(\mathbf{x}_0 + \delta\mathbf{x}, \mathbf{v}_0 + \delta\mathbf{v}) = \mathbf{f}(\mathbf{x}_0, \mathbf{v}_0) + \frac{\partial \mathbf{f}}{\partial \mathbf{x}}\delta\mathbf{x} + \frac{\partial \mathbf{f}}{\partial \mathbf{v}}\delta\mathbf{v}

(derivatives evaluated at (\mathbf{x}_0, \mathbf{v}_0))

Getting specific with \mathbf{f}

\displaystyle \mathbf{f}_i = \frac{1}{m_i}\sum_{j\neq i}k_{ij}\left(\|\mathbf{x}_j - \mathbf{x}_i\| - l_{ij}\right)\frac{\mathbf{x}_j - \mathbf{x}_i}{\|\mathbf{x}_j - \mathbf{x}_i\|}

\displaystyle \mathbf{f}_i = \frac{1}{m_i}\sum_{j\neq i}k_{ij}\left((\mathbf{x}_j - \mathbf{x}_i) - l_{ij} \frac{\mathbf{x}_j - \mathbf{x}_i}{\|\mathbf{x}_j - \mathbf{x}_i\|}\right)

\displaystyle \mathbf{f}_i = \frac{1}{m_i}\sum_{j\neq i}k_{ij} (\mathbf{x}_j - \mathbf{x}_i) - \frac{1}{m_i}\sum_{j\neq i}k_{ij} l_{ij} \frac{\mathbf{x}_j - \mathbf{x}_i}{\|\mathbf{x}_j - \mathbf{x}_i\|}

Deriving against \mathbf{x}_i

\displaystyle \frac{\partial}{\partial \mathbf{x}_i}\left(\frac{1}{m_i}\sum_{j\neq i}k_{ij} (\mathbf{x}_j - \mathbf{x}_i)\right) = \frac{1}{m_i}\sum_{j\neq i}k_{ij} (\mathbf{0} - \mathbf{1}) = -\frac{1}{m_i}\left(\sum_{j\neq i}k_{ij}\right)\mathbf{1}

\displaystyle \left(\frac{\partial}{\partial \mathbf{x}}\frac{\mathbf{y}-\mathbf{x}}{\|\mathbf{y}-\mathbf{x}\|}\right)_{ij} = \frac{\partial}{\partial x_i}\frac{y_j-x_j}{\sqrt{\sum_k (y_k-x_k)^2}}

\displaystyle \left(\frac{\partial}{\partial \mathbf{x}}\frac{\mathbf{y}-\mathbf{x}}{\|\mathbf{y}-\mathbf{x}\|}\right)_{ij} = \frac{\partial}{\partial x_i}\left[(y_j-x_j)\left(\sum_k (y_k-x_k)^2\right)^{-\frac{1}{2}}\right]

\displaystyle \left(\frac{\partial}{\partial \mathbf{x}}\frac{\mathbf{y}-\mathbf{x}}{\|\mathbf{y}-\mathbf{x}\|}\right)_{ij} = -\delta_{ij}\left(\sum_k (y_k-x_k)^2\right)^{-\frac{1}{2}} + (y_j-x_j)\frac{\partial}{\partial x_i}\left(\sum_k (y_k-x_k)^2\right)^{-\frac{1}{2}}

\displaystyle \frac{\partial}{\partial x_i}\left(\sum_k (y_k-x_k)^2\right)^{-\frac{1}{2}} = -\frac{1}{2}\left(\sum_k (y_k-x_k)^2\right)^{-\frac{3}{2}}2\,(y_i-x_i)\,(-1)

\displaystyle \frac{\partial}{\partial x_i}\left(\sum_k (y_k-x_k)^2\right)^{-\frac{1}{2}} = \left(\sum_k (y_k-x_k)^2\right)^{-\frac{3}{2}}(y_i-x_i)

\displaystyle \frac{\partial}{\partial \mathbf{x}}\frac{\mathbf{y}-\mathbf{x}}{\|\mathbf{y}-\mathbf{x}\|} = -\frac{1}{\|\mathbf{y}-\mathbf{x}\|}\mathbf{1} + \frac{1}{\|\mathbf{y}-\mathbf{x}\|^3}(\mathbf{y}-\mathbf{x})(\mathbf{y}-\mathbf{x})^{\rm T}

\displaystyle \frac{\partial}{\partial \mathbf{x}}\frac{\mathbf{y}-\mathbf{x}}{\|\mathbf{y}-\mathbf{x}\|} = \frac{1}{\|\mathbf{y}-\mathbf{x}\|}\left(\mathbf{P_{y-x}} - \mathbf{1}\right)

\displaystyle \frac{\partial}{\partial \mathbf{x}_i}\left(\frac{1}{m_i}\sum_{j\neq i}k_{ij} l_{ij} \frac{\mathbf{x}_j - \mathbf{x}_i}{\|\mathbf{x}_j - \mathbf{x}_i\|}\right) = \frac{1}{m_i}\sum_{j\neq i}\frac{k_{ij} l_{ij}}{\|\mathbf{x}_j-\mathbf{x}_i\|}\left(\mathbf{P}_{\mathbf{x}_j-\mathbf{x}_i} - \mathbf{1}\right)

\displaystyle \frac{\partial}{\partial \mathbf{x}_i}\mathbf{f}_i = -\frac{1}{m_i}\left(\sum_{j\neq i}k_{ij}\right)\mathbf{1} - \frac{1}{m_i}\sum_{j\neq i}\frac{k_{ij} l_{ij}}{\|\mathbf{x}_j-\mathbf{x}_i\|}\left(\mathbf{P}_{\mathbf{x}_j-\mathbf{x}_i} - \mathbf{1}\right)

\displaystyle \frac{\partial}{\partial \mathbf{x}_i}\mathbf{f}_i = \frac{1}{m_i}\left[\sum_{j\neq i}\frac{k_{ij} l_{ij}}{\|\mathbf{x}_j-\mathbf{x}_i\|}\left(\mathbf{1} - \mathbf{P}_{\mathbf{x}_j-\mathbf{x}_i}\right) - \left(\sum_{j\neq i}k_{ij}\right)\mathbf{1}\right]

Deriving against \mathbf{x}_k (k \neq i)

\displaystyle \frac{\partial}{\partial \mathbf{x}_k}\left(\frac{1}{m_i}\sum_{j\neq i}k_{ij} (\mathbf{x}_j - \mathbf{x}_i)\right) = \frac{1}{m_i}\sum_{j\neq i}k_{ij} (\delta_{kj}\mathbf{1} - \delta_{ki}\mathbf{1}) = \frac{k_{ik}}{m_i} \mathbf{1}

\displaystyle \frac{\partial}{\partial \mathbf{x}_k}\left(\frac{1}{m_i}\sum_{j\neq i}k_{ij} l_{ij} \frac{\mathbf{x}_j - \mathbf{x}_i}{\|\mathbf{x}_j - \mathbf{x}_i\|}\right) = \frac{1}{m_i}\sum_{j\neq i}\delta_{kj}\frac{k_{ij} l_{ij}}{\|\mathbf{x}_j-\mathbf{x}_i\|}\left(\mathbf{1}-\mathbf{P}_{\mathbf{x}_j-\mathbf{x}_i}\right)

\displaystyle \frac{\partial}{\partial \mathbf{x}_k}\left(\frac{1}{m_i}\sum_{j\neq i}k_{ij} l_{ij} \frac{\mathbf{x}_j - \mathbf{x}_i}{\|\mathbf{x}_j - \mathbf{x}_i\|}\right) = \frac{1}{m_i}\frac{k_{ik} l_{ik}}{\|\mathbf{x}_k-\mathbf{x}_i\|}\left(\mathbf{1}-\mathbf{P}_{\mathbf{x}_k-\mathbf{x}_i}\right)

\displaystyle \frac{\partial}{\partial \mathbf{x}_k}\mathbf{f}_i = \frac{k_{ik}}{m_i} \mathbf{1} - \frac{1}{m_i}\frac{k_{ik} l_{ik}}{\|\mathbf{x}_k-\mathbf{x}_i\|}\left(\mathbf{1}-\mathbf{P}_{\mathbf{x}_k-\mathbf{x}_i}\right)

\displaystyle \frac{\partial}{\partial \mathbf{x}_k}\mathbf{f}_i = \frac{1}{m_i} \left(k_{ik} - \frac{k_{ik} l_{ik}}{\|\mathbf{x}_k-\mathbf{x}_i\|}\right) \mathbf{1}- \frac{1}{m_i}\frac{k_{ik} l_{ik}}{\|\mathbf{x}_k-\mathbf{x}_i\|}\mathbf{P}_{\mathbf{x}_k-\mathbf{x}_i}

\displaystyle \frac{\partial}{\partial \mathbf{x}_k}\mathbf{f}_i = \frac{k_{ik}}{m_i} \left[\left(1 - \frac{l_{ik}}{\|\mathbf{x}_k-\mathbf{x}_i\|}\right) \mathbf{1} - \frac{l_{ik}}{\|\mathbf{x}_k-\mathbf{x}_i\|}\mathbf{P}_{\mathbf{x}_k-\mathbf{x}_i}\right]

First order expansion of the SMS \mathbf{f}

\displaystyle \mathbf{f}(\mathbf{x}_0 + \delta\mathbf{x}, \mathbf{v}_0 + \delta\mathbf{v}) = \mathbf{f}(\mathbf{x}_0, \mathbf{v}_0) + \mathbf{M}\delta\mathbf{x}

\displaystyle \mathbf{M}_{ii} = \frac{1}{m_i}\left[\sum_{j\neq i}\frac{k_{ij} l_{ij}}{\|\mathbf{x}_j-\mathbf{x}_i\|}\left(\mathbf{1} - \mathbf{P}_{\mathbf{x}_j-\mathbf{x}_i}\right) - \left(\sum_{j\neq i}k_{ij}\right)\mathbf{1}\right]

\displaystyle \mathbf{M}_{ik} = \frac{k_{ik}}{m_i} \left[\left(1 - \frac{l_{ik}}{\|\mathbf{x}_k-\mathbf{x}_i\|}\right) \mathbf{1} - \frac{l_{ik}}{\|\mathbf{x}_k-\mathbf{x}_i\|}\mathbf{P}_{\mathbf{x}_k-\mathbf{x}_i}\right]

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