@@ -169,7 +168,7 @@ This bilinear system (with respect to the virtual displacement field $` \underli
# Resolution of the linear system
## Spatial discretization
In order to solve our previoulsy defined bilinear (with respect to $`\underline{y}`$ and $`\underline{y}^*`$) system, we will be using the standard Galerkin method. It consists of approximating the function of interest (the displacement field in our case) by a finite sum of known shape functions (polynomials usually) $`\phi_k(\underline{\xi})`$ weighted by unkown coefficients $`y_{jk} `$ where $`k`$ is the order of the shape functions used. In 2D, the discretization of the displacement field gives:
In order to solve our previoulsy defined bilinear system (with respect to $`\underline{y}`$ and $`\underline{y}^*`$), we will be using the standard Galerkin method. It consists of approximating the function of interest (the displacement field in our case) by a finite sum of known shape functions (polynomials usually) $`\phi_k(\underline{\xi})`$ weighted by unkown coefficients $`y_{jk} `$ where $`k \in [1, \, N + 1] `$, $` N `$ being the order of the shape functions used. In 2D, the discretization of the displacement field gives:
where $`\underline{\dot{\mathbb{V}}}_h `$ is time derivative of the unkown coefficients $`\dot{y}_{jk}`$ relative to the velocity field (which is itself the time derivative of the unknown weighting coefficients of the displacement field).
This equation can be factorized and simplified as follows:
@@ -261,7 +260,7 @@ where $` \underline{\underline{\mathbb{M}}} `$ corresponds to the <strong>mass m
Here the only time dependency for our elastic problem is the term associated to the inertia, involving the acceleration field $` \ddot{\underline{y}} `$:
This means that in order to solve our system, we just need to update the acceleration values (no need to solve a linear system) with a selected time scheme , once we have the static solution. In this demo, we will implement the Newmark time scheme, which reads:
```math
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@@ -688,7 +687,7 @@ value = 0.04
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##### Transient source.
This corresponds to $` \displaystyle \underline{g}_0(\underline{x}) \quad \text{in} \quad \int_{\Gamma_0^N} \underline{g}_0 \cdot \underline{y}^* \text{d}S_0 `$
This corresponds to $` \displaystyle \underline{g}_0(\underline{x}) \quad \text{in} \quad \int_{\Gamma_0^N} \underline{g}_0 \cdot \underline{y}^* \, \text{d}S_0 `$
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@@ -1079,7 +1078,7 @@ git commit -m "#0 System linear algebra properly initialized; new method RunStat
### Source operator
RHS for the system is just the surfacic source; we therefore need to define the related operator. This corresponds to the surfacic loading vector $` \displaystyle \underline{\mathbb{F}} = \int_{\Gamma_0^N} \underline{\underline{\mathbb{N}}}^T \cdot \underline{g}_0 \text{d}S_0 `$:
RHS for the system is just the surfacic source; we therefore need to define the related operator. This corresponds to the surfacic loading vector $` \displaystyle \underline{\mathbb{F}} = \int_{\Gamma_0^N} \underline{\underline{\mathbb{N}}}^T \cdot \underline{g}_0 \, \text{d}S_0 `$:
We also need to define the sitffness matrix. This corresponds to $` \displaystyle \underline{\underline{\mathbb{K}}} = \int_{\Omega_{0}}^{} \underline{\underline{\mathbb{B}}}^T \cdot \underline{\underline{\hat{\text{A}}}} \cdot \underline{\underline{\mathbb{B}}} \textrm{d}\Omega_0 `$
We also need to define the sitffness matrix. This corresponds to $` \displaystyle \underline{\underline{\mathbb{K}}} = \int_{\Omega_{0}}^{} \underline{\underline{\mathbb{B}}}^T \cdot \underline{\underline{\hat{\text{A}}}} \cdot \underline{\underline{\mathbb{B}}} \, \textrm{d}\Omega_0 `$
@@ -1375,7 +1374,7 @@ git commit -m "#0 Static case implemented and working."
## Defining the mass operator
Here we are defining the mass matrix required for the dynamic part of the run $` \underline{\underline{\mathbb{M}}} = \int_{\Omega_0} \rho_0 \underline{\underline{\mathbb{N}}} ^T \cdot \underline{\underline{\mathbb{N}}} \text{d}\Omega_0 `$
Here we are defining the mass matrix required for the dynamic part of the run $` \underline{\underline{\mathbb{M}}} = \int_{\Omega_0} \rho_0 \underline{\underline{\mathbb{N}}} ^T \cdot \underline{\underline{\mathbb{N}}} \, \text{d}\Omega_0 `$