# 4D Remeshing Using a Space-Time Finite Element Method for Elastodynamics Problems

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## Abstract

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## 1. Introduction

## 2. Principle of the Method

- Choosing a Lagrange interpolation for displacements implies that displacements are continuous, but the velocities are discontinuous. As a consequence, integration by parts as in (3) is not totally rigorous, and it could be necessary to use the discontinuous Galerkin formulation, which amounts to writing the derivative of velocity within the theory of distributions. We will preserve the formulation in (3), knowing that the error here is of the same order as in the case of traditional finite elements in space. Indeed, with a Lagrange interpolation local displacements are continuous, whereas the global deformation is discontinuous.
- Even if it is not absolutely necessary, the advantage of using a laminated mesh such as defined here is that it becomes possible, rather than assembling the total matrix $\left[T\right]$, to only assemble the sub-matrices $\left[{T}_{ij}\right]$. This considerably reduces the size of the systems to be solved. More precisely, the size of these linear systems is exactly the same as that obtained in the case of approaches based on the coupling of finite incremental differences in time with finite elements in space. Moreover, the method is not limited to simplex elements, and the spatial position of each set of nodes can vary from one time plane to the other. It is one of the main advantages of the method.
- We specify that the nodal vector relating the boundary conditions with velocity $\{\Lambda \}$ is written as:$$\{\Lambda \}={\left(\left\{{\Lambda}_{0}\right\},0,...,0,\left\{{\Lambda}_{n}\right\}\right)}^{T}$$The first system of equations,$$\left[{T}_{11}\right]\left\{{U}_{0}\right\}+\left[{T}_{12}\right]\left\{{U}_{1}\right\}=\left\{{F}_{0}\right\}+\left\{{\Lambda}_{0}\right\}$$$$\left[{T}_{i/i-1}\right]\left\{{U}_{i-2}\right\}+\left[{T}_{i/i}\right]\left\{{U}_{i-1}\right\}+\left[{T}_{i/i+1}\right]\left\{{U}_{i}\right\}=\left\{{F}_{i-1}\right\}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}2\le i\le n\phantom{\rule{0.166667em}{0ex}},$$$$\left[{T}_{n+1/n}\right]\left\{{U}_{n-1}\right\}+\left[{T}_{n+1/n+1}\right]\left\{{U}_{n}\right\}=\left\{{F}_{n}\right\}+\left\{{\Lambda}_{n}\right\}\phantom{\rule{0.166667em}{0ex}},$$
- Finally, the matrices of resolution $\left[{T}_{i/i+1}\right]$ are generally non-symmetric, even if the total matrix $\left[T\right]$ is symmetric. Thus, for the algorithm presented above, a non-symmetrical solver should be used. This can appear penalizing in terms of computing time. However, since the final objective is to use this approach to deal with problems of contact with friction and since the nonlinear resolution we developed in [14] is of the Gauss–Seidel nonlinear type, asymmetries do not affect computing time.

## 3. 4D Mesh and Remeshing

#### 3.1. 4D Mesh Generation

#### 3.2. Remeshing Technique

## 4. Numerical Analysis

#### 4.1. Stability

- For 1D space-time elastodynamic applications, the use of the STFEM method with linear simplex elements is similar to the use of the implicit Newmark integration scheme with $\delta =1/2$ and $\theta =1/3$. The method is then unconditionally stable.
- For 2D space-time elastodynamic applications, the use of the STFEM method with linear simplex elements is similar to the use of the explicit Newmark integration scheme with $\delta =1/2$ and $\theta =0$. The method is then conditionally stable. Classically, the time step has to verify the CFLcondition: $\Delta t\le {min}_{J}\frac{2}{{\omega}_{J}}$, where each ${\omega}_{J}$ is the frequency of a normal mode of vibration.
- For higher dimensions (3D and 4D), no direct relationship between the STFEM and the Newmark method has been established. Nevertheless, we noted that our method required sufficiently small space-time slabs, of the same order of the discretization time step necessary with explicit methods of integration.
- Furthermore, the use of the STFEM method with multiplex elements is similar to the use of the implicit Newmark integration scheme with $\delta =1/2$ and $\theta =1/3$, for 1D, 2D, 3D and 4D space-time applications. In this case, the method is unconditionally stable.

#### 4.2. Convergence

## 5. Numerical Results on Mesh Adaptation

## 6. Conclusions

## Author Contributions

## Conflicts of Interest

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**Figure 4.**3D initial mesh (scheme at the left); part of the 4D mesh generated by the triangle (1; 2; 4) common to the two finite elements of the 3D initial mesh (scheme at the right).

**Figure 5.**Space-time mesh generation by rotation of the loaded area. The loaded area is represented by arrows.

**Figure 7.**Size of the time step of discretization $\Delta t$ necessary for stability with respect to the average size h of the 3D finite elements.

**Figure 8.**Maximum error over the space, at the last time step, between the analytic solution and the numerical solution for each mesh size h, in the logarithmic scale.

**Figure 11.**Comparison of vertical displacements for points situated on the circle of control, expressed in mm, at time $t=2.{10}^{-5}$ s. Each check point is defined by its angular coordinate, expressed in radians.

**Figure 12.**Comparison of vertical displacements for points situated on the circle of control, expressed in mm, at time $t=4.{10}^{-5}$ s. Each check point is defined by its angular coordinate, expressed in radians.

**Figure 13.**Isovalues of the norm of nodal displacements, expressed in mm, at $t=2.{10}^{-5}$ s for the coarse mesh (image at the top) and for the fine mesh (image at the bottom).

**Figure 14.**Isovalues of the norm of nodal displacements, expressed in mm, at $t=4.{10}^{-5}$ s for the coarse mesh (image on the top) and for the fine mesh (image on the bottom).

**Table 1.**Table of connectivities of the 4D space-time mesh resulting from the elementary 3D space mesh of Figure 4.

Element Number | Node 1 | Node 2 | Node 3 | Node 4 | Node 5 |
---|---|---|---|---|---|

1 | 1 | 2 | 3 | 4 | 6 |

2 | 2 | 3 | 4 | 6 | 7 |

3 | 3 | 4 | 6 | 7 | 8 |

4 | 4 | 6 | 7 | 8 | 9 |

5 | 1 | 2 | 4 | 5 | 6 |

6 | 2 | 4 | 5 | 6 | 7 |

7 | 4 | 5 | 6 | 7 | 9 |

8 | 5 | 6 | 7 | 9 | 10 |

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**MDPI and ACS Style**

Dumont, S.; Jourdan, F.; Madani, T. 4D Remeshing Using a Space-Time Finite Element Method for Elastodynamics Problems. *Math. Comput. Appl.* **2018**, *23*, 29.
https://doi.org/10.3390/mca23020029

**AMA Style**

Dumont S, Jourdan F, Madani T. 4D Remeshing Using a Space-Time Finite Element Method for Elastodynamics Problems. *Mathematical and Computational Applications*. 2018; 23(2):29.
https://doi.org/10.3390/mca23020029

**Chicago/Turabian Style**

Dumont, Serge, Franck Jourdan, and Tarik Madani. 2018. "4D Remeshing Using a Space-Time Finite Element Method for Elastodynamics Problems" *Mathematical and Computational Applications* 23, no. 2: 29.
https://doi.org/10.3390/mca23020029