# An Improved Material Point Method with Aggregated and Smoothed Bernstein Functions

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## Abstract

**:**

## 1. Introduction

## 2. The Material Point Method

#### 2.1. Governing Equations

#### 2.2. Weak Form

#### 2.3. Spatial Discretization

#### 2.4. A Typical Explicit Workflow

## 3. Improvement Strategy

#### 3.1. Basic Requirements of Shape Functions

- (1)
- The partition of unity (PU), ${{\displaystyle \sum}}_{I=1}^{{n}_{n}}{S}_{I}\left(x\right)=1$ for all $x$.
- (2)
- The compact support (CS), ${S}_{I}\left(x\right)\ne 0$ for locations close enough to node $I$.
- (3)
- The non-negativity (NN), ${S}_{I}\left(x\right)\ge 0$ for all $x$.

#### 3.2. Bernstein Polynomials

#### 3.3. Smooth Transform for Avoiding Cell Crossing Errors

#### 3.4. Aggregation Transform for Reducing Node Amount

**ASBMPM.**It should be noted that the ASB functions meet the partition of unity, compact support, and non-negativity.

## 4. Formulation Summary

## 5. Numerical Validation

#### 5.1. Axial Vibration of a 1D Continuum Bar

#### 5.1.1. Resistance to Cell Crossing Errors

#### Small Displacement Case (${v}_{0}=0.1\mathrm{m}/\mathrm{s}$, Duration = 50 s)

#### Large Displacement Case (${v}_{0}=0.75\mathrm{m}/\mathrm{s}$, Duration = 50 s)

#### 5.1.2. Reduction of Numerical Fracture

#### 5.1.3. Spatial Convergence

#### 5.2. Impact of Two 2D Elastic Disks

#### 5.2.1. System Energy Conversation

#### 5.2.2. Contact Feature

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**Typical aggregated-smoothed Bernstein shape functions: (

**a**) the quadratic, (

**b**) the cubic. Quadratic B-spline and cubic B-spline are exactly the lowest-level cases of the ASB hierarchy.

**Figure 6.**D continuum bar vibration results (the free-end particle) for variations of the MPM algorithm under the initial velocity amplitude ${v}_{0}=0.1\mathrm{m}/\mathrm{s}$: (

**a**) the displacement, (

**b**) the relative error of displacement against the analytic solution, (

**c**) the velocity, and (

**d**) the relative error of velocity against the analytic solution.

**Figure 7.**D continuum bar vibration results (the free-end particle) for variations of the MPM algorithm under the initial velocity amplitude ${v}_{0}=0.75\mathrm{m}/\mathrm{s}$: (

**a**) the relative error of displacement against the analytic solution, and (

**b**) the relative velocity error against the analytic solution. For clarity, some series are truncated when their relative errors increase beyond 5.0%.

**Figure 8.**D continuum bar vibration (the free-end particle) for variations of the MPM algorithm: (

**a**) the relative error of displacement against the analytic solution under ${v}_{0}=1.0\mathrm{m}/\mathrm{s}$, (

**b**) the relative error of displacement against the analytic solution under ${v}_{0}=2.0\mathrm{m}/\mathrm{s}$. For clarity, some series are truncated when their relative error increases beyond 5.0%.

**Figure 9.**Maximum durations for MPM variations to maintain 1D continuum bar vibration within 5% error.

**Figure 12.**System energy comparison of 2D disks’ collision problem under different MPM shape functions.

**Figure 13.**Impact comparison of 2D disks’ collision problem under different MPM shape functions: (

**a**) cubic B-spline, (

**b**) cubic-Ⅲ ASB, (

**c**) quadratic B-spline, (

**d**) quadratic-Ⅲ ASB, (

**e**) quadratic-Ⅴ ASB, (

**f**) quadratic-Ⅶ ASB, (

**g**) cpGIMP, (

**h**) uGIMP. The moment t = 2.0 s when disks deform the most is chosen. Each particle is colored according to the volume ratio $J$ to reflect deformation distribution.

**Figure 14.**Departure comparison of 2D disks’ collision problem under different MPM shape functions: (

**a**) cubic B-spline, (

**b**) cubic-III ASB, (

**c**) quadratic B-spline, (

**d**) quadratic-III ASB, (

**e**) quadratic-V ASB, (

**f**) quadratic-VII ASB, (

**g**) cpGIMP, (

**h**) uGIMP. The moment t = 3.0 s is chosen. Each particle is colored according to the volume ratio $J$ to reflect deformation distribution.

Original Bernstein Degree | Continuity after Aggregation and Smoothing | |
---|---|---|

Quadratic (after Once Smoothing) | Cubic (after Twice Smoothing) | |

I | $\left\{\begin{array}{cc}\frac{3}{4}-{r}^{2}& 0\le r<\frac{1}{2}\hfill \\ \frac{1}{8}{(3-2r)}^{2}& \frac{1}{2}\le r<\frac{3}{2}\hfill \\ 0& otherwise\hfill \end{array}\right.$ | $\left\{\begin{array}{cc}\frac{1}{6}\left(3\left(r-2\right){r}^{2}+4\right)& 0\le r<1\hfill \\ -\frac{1}{6}{(r-2)}^{3}& 1\le r<2\hfill \\ 0& otherwise\hfill \end{array}\right.$ |

II | same as above | same as above |

III | $\left\{\begin{array}{cc}{r}^{4}-\frac{3{r}^{2}}{2}+\frac{13}{16}& 0\le r<\frac{1}{2}\hfill \\ -\frac{1}{32}{(2r-3)}^{3}\left(2r+1\right)& \frac{1}{2}\le r<\frac{3}{2}\hfill \\ 0& otherwise\hfill \end{array}\right.$ | $\left\{\begin{array}{cc}\frac{1}{20}\left(-6{r}^{5}+15{r}^{4}-20{r}^{2}+14\right)& 0\le r<1\hfill \\ \frac{1}{20}{(r-2)}^{4}\left(2r+1\right)& 1\le r<2\hfill \\ 0& otherwise\hfill \end{array}\right.$ |

IV | same as above | same as above |

V | $\left\{\begin{array}{cc}\frac{1}{32}\left(-64{r}^{6}+80{r}^{4}-60{r}^{2}+27\right)& 0\le r<\frac{1}{2}\\ \frac{1}{64}{(3-2r)}^{4}\left(4{r}^{2}+1\right)& \frac{1}{2}\le r<\frac{3}{2}\\ 0& otherwise\end{array}\right.\begin{array}{cc}& \end{array}$ | $\begin{array}{cc}\left\{\begin{array}{cc}\frac{3{r}^{7}}{7}-\frac{3{r}^{6}}{2}+\frac{3{r}^{5}}{2}-{r}^{2}+\frac{5}{7}& 0\le r<1\\ -\frac{1}{14}{(r-2)}^{5}\left(r\left(2r-1\right)+1\right)& 1\le r<2\\ 0& otherwise\end{array}\right.& \end{array}$ |

VI | same as above | same as above |

VII | $\begin{array}{cc}\left\{\begin{array}{cc}5{r}^{8}-7{r}^{6}+\frac{35{r}^{4}}{8}-\frac{35{r}^{2}}{16}+\frac{221}{256}& 0\le r<\frac{1}{2}\\ -\frac{1}{512}{(2r-3)}^{5}\left(2r\left(10r\left(2r-1\right)+7\right)+1\right)& \frac{1}{2}\le r<\frac{3}{2}\\ 0& otherwise\end{array}\right.& \end{array}$ | $\begin{array}{cc}\left\{\begin{array}{cc}-\frac{5{r}^{9}}{6}+\frac{15{r}^{8}}{4}-6{r}^{7}+\frac{7{r}^{6}}{2}-{r}^{2}+\frac{13}{18}& 0\le r<1\\ \frac{1}{36}{(r-2)}^{6}\left(r\left(5r\left(2r-3\right)+12\right)-2\right)& 1\le r<2\\ 0& otherwise\end{array}\right.& \end{array}$ |

$\mathrm{where}r=\frac{\left|x-{x}_{I}\right|}{h}$. |

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## Share and Cite

**MDPI and ACS Style**

Zhu, Z.; Bao, T.; Zhu, X.; Gong, J.; Hu, Y.; Zhang, J.
An Improved Material Point Method with Aggregated and Smoothed Bernstein Functions. *Mathematics* **2023**, *11*, 907.
https://doi.org/10.3390/math11040907

**AMA Style**

Zhu Z, Bao T, Zhu X, Gong J, Hu Y, Zhang J.
An Improved Material Point Method with Aggregated and Smoothed Bernstein Functions. *Mathematics*. 2023; 11(4):907.
https://doi.org/10.3390/math11040907

**Chicago/Turabian Style**

Zhu, Zheng, Tengfei Bao, Xi Zhu, Jian Gong, Yuhan Hu, and Jingying Zhang.
2023. "An Improved Material Point Method with Aggregated and Smoothed Bernstein Functions" *Mathematics* 11, no. 4: 907.
https://doi.org/10.3390/math11040907