# An RBF Meshless Approach to Evaluate Strain Due to Large Displacements in Flexible Printed Circuit Boards

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Background

#### 2.1. Radial Basis Functions

#### 2.2. Kinematics of Large Displacements for a Bi-Dimensional Plate

#### 2.3. Analytical Test Cases

#### 2.3.1. Test Case 1

#### 2.3.2. Test Case 2

## 3. Results

#### 3.1. Test Case 1

#### 3.2. Test Case 2

#### 3.3. CAD-Based Test Case

#### 3.4. Result

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Example of flexible-PCB study [15] (

**a**) real flexible PCB in mounting configuration, (

**b**) FEM model in un uninstalled state with trace mapping and (

**c**) FEM model in installed state with and without trace mapping.

**Figure 3.**(

**a**) Test-case geometry with rotated cylinder and (

**b**) displacement field imposed to RBF centers.

**Figure 4.**Left: RBF membrane deformations along the x- (

**a**) and y-axes (

**c**). Right: APDL membrane deformations along the x- (

**b**) and y-axes (

**d**).

**Figure 6.**${\u03f5}_{x}$ and ${\u03f5}_{y}$ comparison on the top surface for the RBF and FEM models. From left to right: upper row (

**a**,

**b**), RBF model; bottom row (

**c**,

**d**), FEM model.

**Figure 7.**Comparison of in-plane shear deformations for the RBF (

**a**) and FEM (

**b**) models on the top surface.

**Figure 11.**Top strain for the proposed method (

**a**,

**b**) and FEM (

**c**,

**d**) for an increasing number of points along the x-direction.

**Figure 12.**Convergence of the maximum strain for both the FEM and RBFLD methods with the increase in the number of points.

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**MDPI and ACS Style**

Groth, C.; Chiappa, A.; Porziani, S.; Salvini, P.; Biancolini, M.E.
An RBF Meshless Approach to Evaluate Strain Due to Large Displacements in Flexible Printed Circuit Boards. *Micromachines* **2022**, *13*, 1163.
https://doi.org/10.3390/mi13081163

**AMA Style**

Groth C, Chiappa A, Porziani S, Salvini P, Biancolini ME.
An RBF Meshless Approach to Evaluate Strain Due to Large Displacements in Flexible Printed Circuit Boards. *Micromachines*. 2022; 13(8):1163.
https://doi.org/10.3390/mi13081163

**Chicago/Turabian Style**

Groth, Corrado, Andrea Chiappa, Stefano Porziani, Pietro Salvini, and Marco Evangelos Biancolini.
2022. "An RBF Meshless Approach to Evaluate Strain Due to Large Displacements in Flexible Printed Circuit Boards" *Micromachines* 13, no. 8: 1163.
https://doi.org/10.3390/mi13081163