# Second-Order Collocation-Based Mixed FEM for Flexoelectric Solids

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Flexoelectricity: Constitutive and Governing Equations

## 3. Collocation Mixed Finite Element Method (CMFEM)

#### 3.1. CMFEM for Linear Element

Algorithm 1 Collocation method. | ||

1: | compute all ${\mathbf{J}}^{-1}$ | ▹ The inverse Jacobian matrix for all integration points (ip) |

2: | ||

3: | compute ${\mathbf{B}}_{\mathbf{u},N}$ according to Equation (27) | ▹${\mathbf{B}}_{\mathbf{u},N}$ and ${\mathbf{B}}_{\varphi ,N}$ contain the B-Matrices derived |

4: | compute ${\mathbf{B}}_{\varphi ,N}$ according to Equation (28) | ▹ from nodal values N for all collocation points |

5: | ||

6: | define ${\mathbf{A}}^{-1}$ | ▹ containing precalculated values of $\mathbf{A}$ according to Equation (37) |

7: | ||

8: | for 1,…,${n}_{ip}$ do | ▹ Loop over all ip |

9: | ||

10: | compute $\mathbf{P}$ according to Equation (32) | ▹ The P-vector for current ip |

11: | $\mathbf{P}$ = (1,${r}_{ip}$,${s}_{ip}$,${r}_{ip}{s}_{ip}$) | |

12: | ||

13: | compute ${\mathbf{P}}_{,1}$ and ${\mathbf{P}}_{,3}$ Equation (44) | ▹ The directional derivatives |

14: | ▹ of the P-vector for current ip | |

15: | ${\mathbf{P}}_{,1}$ = (0,${j}_{11}^{inv}$,${j}_{13}^{inv}$,${j}_{11}^{inv}{s}_{ip}$+${j}_{13}^{inv}{r}_{ip}$) | |

16: | ${\mathbf{P}}_{,3}$ = (0,${j}_{31}^{inv}$,${j}_{33}^{inv}$,${j}_{31}^{inv}{s}_{ip}$+${j}_{33}^{inv}{r}_{ip}$) | ▹ Use inverse Jacobian of current ip |

17: | ||

18: | compute ${\mathbf{B}}_{\mathbf{u}}$ according to Equation (41) | ▹ The collocated ${\mathbf{B}}_{\mathbf{u}}$-matrix |

19: | ||

20: | compute ${\mathbf{B}}_{\nabla \mathbf{u}}$ according to Equation (42) | ▹ The directional derivative of ${\mathbf{B}}_{\mathbf{u}}$-matrix |

21: | ||

22: | compute ${\mathbf{B}}_{\varphi}$ according to Equation (41) | ▹ The collocated ${\mathbf{B}}_{\varphi}$-matrix |

23: | ||

24: | compute ${\mathbf{B}}_{\nabla \varphi}$ according to Equation (43) | ▹ The directional derivative of ${\mathbf{B}}_{\varphi}$-matrix |

25: | ||

26: | … | |

27: | end for |

#### 3.2. CMFEM for Quadratic Element

**A**-matrix changes to $\mathbf{A}={\mathbf{A}}_{quad}$ with ${\mathbf{A}}_{quad}$ as

## 4. Numerical Examples

#### 4.1. Cantilever Beam Problem

#### 4.1.1. Verification of Linear Element

#### 4.1.2. Comparison of Linear and Quadratic Elements

#### 4.1.3. Convergence Study

#### 4.1.4. Investigation of Differences between Linear and Quadratic Elements

#### 4.1.5. Parameter Study of Flexoelectric Coefficients

#### 4.1.6. Discussion about the Advantages of Quadratic Element

#### 4.2. Truncated Pyramid Compression Problem

## 5. Conclusions

- A linear CMFEM element is developed and verified with numerical results for a 2D cantilever beam problem from the literature.
- The correctness of the newly proposed quadratic element is verified through comparison with the linear element and examples from the literature. To make the closest comparison, the strain gradient components, which cannot be computed by the linear element, are hardcoded to zero in the quadratic UEL.
- Based on a convergence study, it is shown that the quadratic element is more accurate than the linear element. In particular, for meshes with very few elements, the accuracy is drastically improved with the usage of the newly developed quadratic element.
- By no longer hardcoding specific strain gradient components to zero in the quadratic UEL, CMFEM elements’ performance is analyzed. There, it is presented that mechanical strain gradient ${\eta}_{333}$ cannot be captured by the linear element, but has a relevant magnitude, leading to a strong overestimation of flexoelectric coupling and mechanical stiffness.
- The abovementioned findings are further investigated by a parameter study in which cantilever beam’s deflection and generated electric field are studied for a wide range of flexoelectric coefficients. There, it is found that using the linear element, the cantilever beams’ deflection nears zero for high flexoelectric coefficients and the remaining deflection is build-up only from shear stress. For the quadratic element, on the other hand, the deflection saturates at a nonzero value mimicking a physically realistic behavior.
- In the flexoelectric truncated pyramid example, although the linear element produced physically invalid results, the results of the quadratic element are in a very good agreement with the literature. The newly proposed quadratic element yields realistic results compared to those obtained in experiments [8,34] and a distribution of the electric field similar to [12,16]. This is additionally outlined in a simulation series wherein the flexoelectric size effect is visualized.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

BVP | boundary value problem |

CMFEM | collocation-based mixed finite element method |

cw | current work |

FEM | finite element method |

MEMS | microelectromechanical systems |

SGE | strain gradient elasticity |

UEL | user element |

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**Figure 1.**General two-dimensional coordinate transformation of a quadrilateral element from local curvilinear coordinates to a global Cartesian coordinate system.

**Figure 2.**Four-noded quadrilateral element in the local curvilinear coordinate system with 2 × 2 Gaussian quadrature points.

**Figure 3.**The nine-noded quadrilateral element in the local curvilinear coordinate system with 3 × 3 Gaussian quadrature points.

**Figure 5.**Deflections ${u}_{3}$ and ${u}_{1}$, mechanical strain ${\epsilon}_{11}$, and mechanical strain gradient ${\eta}_{311}$ distribution along the cantilever beam.

**Figure 6.**Comparison of deflection of the cantilever beam between the results in the literature and of the current work using linear element for different flexoelectric coefficients ${f}_{1}$. “cw” denotes the current work.

**Figure 7.**Electric potential $\varphi $ (top) and electric field ${E}_{3}$ (bottom) distribution in the cantilever beam.

**Figure 8.**Comparison of electric field of the cantilever beam between the results given in the literature and of the current work by using linear element for different flexoelectric coefficients ${f}_{1}$. cw denotes the current work.

**Figure 9.**Deflection ${u}_{3}$ of the cantilever beam for linear and quadratic element with neglected strain gradient components, along with analytical solution. cw denotes the current work.

**Figure 10.**Electric field ${E}_{3}$ generated in the cantilever beam for linear and quadratic element with neglected strain gradient components vs. analytical solution. cw denotes the current work.

**Figure 11.**Coarse (top) and fine (bottom) mesh used for the convergence analysis of the cantilever beam problem with total element number n. Not all elements are shown for better visualization.

**Figure 12.**Relative errors and convergence rates of displacement ${u}_{3}$ and electric field ${E}_{3}$ for the developed linear and quadratic element with neglected strain gradient components.

**Figure 13.**Comparison of deflection calculated by using linear and quadratic element along the length of the beam.

**Figure 14.**Comparison of electric field calculated by using linear and quadratic element along the length of the beam.

**Figure 15.**Comparison of the mechanical strain components calculated by using linear and quadratic element through the height of the beam for prescribed mechanical displacement.

**Figure 16.**Comparison of mechanical stress components calculated by using linear and quadratic element along the height of the beam for prescribed mechanical displacement.

**Figure 17.**Comparison of mechanical strain gradient components calculated by using linear and quadratic element along the length of the beam for prescribed mechanical displacement.

**Figure 19.**Comparison of mechanical strain gradient ${\eta}_{333}$ distribution for the linear (top) and quadratic (bottom) element along the cantilever beam.

**Figure 20.**Comparison of higher-order stress components calculated by using linear and quadratic element along the length of the beam for prescribed mechanical displacement.

**Figure 21.**Flexoelectric and mechanical strain gradient contributions to the higher-order stress components ${\tau}_{311}$ and ${\tau}_{333}$ for linear and quadratic element along the length of the beam for prescribed mechanical displacement.

**Figure 22.**Comparison of electric field ${E}_{3}$ distribution based on linear (top) and quadratic (bottom) element in the cantilever beam.

**Figure 23.**Comparison of electric displacement components calculated by using linear and quadratic element along the height of the beam for prescribed mechanical displacement.

**Figure 24.**Maximum deflection ${u}_{3}$ of the cantilever beam calculated by using linear and quadratic element for various flexoelectric coefficients.

**Figure 25.**Maximum electric field ${E}_{3}$ of the cantilever beam calculated by using quadratic element for various flexoelectric coefficients.

**Figure 26.**Comparison of cantilever beam bending for a small and high flexoelectric coefficient by using the linear and quadratic element.

**Figure 27.**Discontinuous strain ${\epsilon}_{33}$ for linear and quadratic element across the height of the cantilever beam.

**Figure 29.**Contour plots for mechanical strain ${\epsilon}_{33}$ and electric potential $\varphi $ in the truncated pyramid using linear and quadratic elements. The same color scale is used for both elements for better comparison.

**Figure 31.**Effective electric field ${E}_{\mathrm{eff}}$ of the truncated pyramid for linear element, quadratic element and [16] for different geometric sizes.

**Table 1.**Boundary conditions and loads for the cantilever beam [25].

Side/Corner | Mechanical BC | Electrical BC | Prescribed Force |
---|---|---|---|

AB | ${u}_{1}=0$ | - | - |

A | ${u}_{3}=0$ | - | - |

CD | - | $\varphi =0$ | $F=1$ nN in ${x}_{3}$ |

**Table 2.**Material properties for the cantilever beam [25].

E [GPa] | $\mathit{\nu}$ [-] | l [nm] | ${\mathit{a}}_{1}$, ${\mathit{a}}_{3}\left[\frac{{\mathbf{C}}^{2}}{{\mathbf{Nm}}^{2}}\right]$ | ${\mathit{f}}_{1}\left[\frac{\mathit{\mu}\mathbf{C}}{\mathbf{m}}\right]$ | ${\mathit{f}}_{2}\left[\frac{\mathit{\mu}\mathbf{C}}{\mathbf{m}}\right]$ |
---|---|---|---|---|---|

126 | 0 | 2 | $13.0\times {10}^{-9}$ | $0.1$ | 0 |

Side | Mechanical BC | Electrical BC | Prescribed Force |
---|---|---|---|

AD | ${u}_{1}={u}_{3}=0$ | $\varphi =V=\mathrm{const}$ | - |

BC | - | $\varphi =0$ | $F=4.5\frac{\mathrm{N}}{\mathrm{mm}}$ in ${x}_{3}$ |

**Table 4.**Material properties for the truncated pyramid [16].

E [GPa] | $\mathit{\nu}$ [-] | l [nm] | ${\mathit{a}}_{1}$, ${\mathit{a}}_{3}\left[\frac{{\mathbf{C}}^{2}}{{\mathbf{Nm}}^{2}}\right]$ | ${\mathit{f}}_{1}\left[\frac{\mathit{\mu}\mathbf{C}}{\mathbf{m}}\right]$ | ${\mathit{f}}_{2}\left[\frac{\mathit{\mu}\mathbf{C}}{\mathbf{m}}\right]$ |
---|---|---|---|---|---|

100 | $0.37$ | 0 | $11.0\times {10}^{-9}$ | $1.0$ | 0 |

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**MDPI and ACS Style**

Tannhäuser, K.; Serrao, P.H.; Kozinov, S.
Second-Order Collocation-Based Mixed FEM for Flexoelectric Solids. *Solids* **2023**, *4*, 39-70.
https://doi.org/10.3390/solids4010004

**AMA Style**

Tannhäuser K, Serrao PH, Kozinov S.
Second-Order Collocation-Based Mixed FEM for Flexoelectric Solids. *Solids*. 2023; 4(1):39-70.
https://doi.org/10.3390/solids4010004

**Chicago/Turabian Style**

Tannhäuser, Kevin, Prince Henry Serrao, and Sergey Kozinov.
2023. "Second-Order Collocation-Based Mixed FEM for Flexoelectric Solids" *Solids* 4, no. 1: 39-70.
https://doi.org/10.3390/solids4010004