# A Novel Calculation Method of Hydrodynamic Pressure Based on Polyhedron SBFEM and Its Application in Nonlinear Cross-Scale CFRD-Reservoir Systems

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. A Calculation Method of Hydrodynamic Pressure and Polyhedral Fluid Element

#### 2.1. Computation Method of Hydrodynamic Pressure Based on Polyhedron SBFEM

^{2}p = 0

_{0}boundary condition of reservoir water is:

_{1}of dam satisfies:

_{n}

_{2}between the reservoir and the river–valley satisfies:

_{n}

^{2}is the Laplacian operator, p is the hydrodynamic pressure, n is the normal direction of the interface, ρ represents the fluid density, and ü

_{n}and ϋ

_{n}are the normal accelerations of the dam–reservoir interface and the river–valley interface, respectively. Since the whole semi-infinite reservoir water in front of the dam is discretized by SBFEM, the radiation boundary condition at infinity S

_{3}of the reservoir is satisfied automatically, the theory of which is expounded as below.

_{1}is [0, +∞] from upstream face of the dam to infinity of the reservoir. Furthermore, the range of local circumferential coordinates ξ

_{2}and ξ

_{3}is [−1, +1]. By using the scaled boundary coordinate transformation, the coordinates of the global Cartesian coordinate system at any point (X

_{1}, X

_{2}, X

_{3}) in the reservoir can be expressed by the local scaled boundary coordinates (ξ

_{1}, ξ

_{2}, ξ

_{3}). Here, ξ

_{1}serves as a factor of proportionality, as follows:

_{1}, x

_{2}, x

_{3}) represent global coordinates of a node on a reservoir grid at the dam upstream face (ξ

_{1}= 0), and $\left[N\left({\xi}_{2},{\xi}_{3}\right)\right]$ denotes the polygon mean-value shape function, which is compatible with the polygon mesh and particularly presented in Section 2.2 below. With the help of the interpolation function $\left[N\left({\xi}_{2},{\xi}_{3}\right)\right]$, the hydrodynamic pressure p(ξ

_{1}, ξ

_{2}, ξ

_{3}) at any point in a polygon element can be expressed by the hydrodynamic pressure {p(ξ

_{1})} at nodes of the fluid element as:

^{−1}= [{b

^{1}} {b

^{2}} {b

^{3}}].

^{0}], [E

^{1}], [E

^{2}], [C

^{0}], and [M

^{1}] only depend on geometry information of the mesh on the upstream face of the dam and are expressed as follows:

_{2}on the (X

_{2}, X

_{3}) plane.

_{i}] is the diagonal matrix, and the real part of λ

_{i}≥ 0.

_{n}} and that caused by the vibration of the river valley {ϋ

_{n}} in the reservoir.

#### 2.2. Polyhedral Scaled Boundary Finite Element of Fluid

#### 2.2.1. Polygon Mean-Value Shape Function

_{i}(in Figure 4). Point M is selected as the geometric center of the polygon, and n represents the number of vertices on the polygon, that is, the number of edges of the polygon. The interpolation function expressed in Equations (30)–(32) can be used for both convex and concave polygons.

_{1}, X

_{2}) ∈ W

_{e}. In order to simplify the integration of the element matrix, it is necessary to construct a conforming approximation of the polygon using mean-value shape functions. Similar to the isoparametric element in FEM, the mean shape function is defined on a standard element in the local coordinate system (ξ

_{1}, ξ

_{2}) ∈ W

_{0}. Four standard elements in the local coordinate system are illustrated as an example in Figure 5: a regular triangle, quadrilateral, pentagon, and hexagon. The vertices of each standard element are placed on the same circumscribed circle with a radius of 1.0, and the geometric center of each element coincides with the center of the circumscribed circle. Therefore, any point in a standard polygon element can be directly connected with each vertex without being occluded. The coordinates of the polygon vertices on the unit circle are (cos 2π/n, sin 2π/n), (cos 4π/n, sin 4π/n), …, and (1, 0), where n is the number of vertices. In this way, the shape function can be defined and used in the local coordinate system as shown in Figure 5, where only four kinds of polygons are shown, and any polygon in the global Cartesian coordinates can be transformed into a standard element using the corresponding local coordinate system through the polygon isoparametric mapping shape function

**F**. An example of the mapping process for a pentagon is illustrated in Figure 6.

^{0}], [E

^{1}], [E

^{2}], [M

^{1}], and so on, are computed by integrating over the triangular subunits using the standard orthogonal criterion. The triangular subunits in Figure 7 are used only for numerical integration. A detailed discussion of the integration methods can be found in references [29,52].

#### 2.2.2. Polyhedral Fluid Elements

**F**. Subsequently, the matrices ([E

^{0}], [E

^{1}], [E

^{2}], [M

^{1}], and so on) for the polyhedral element are integrated and calculated. In the end, the total matrices of all reservoir elements are integrated, and the hydrodynamic pressure can be directly solved as mentioned in Section 2.1.

## 3. A Nonlinear Dynamic Coupling Method for Cross-Scale Dam-Reservoir Systems Based on the Polyhedron SBFEM

#### 3.1. Polyhedron SBFEM Procedure for Fluid

#### 3.2. Nonlinear Dynamic Coupling Method for Cross-Scale CFRD-Reservoir Systems

_{s}], [C

_{s}], and [K

_{s}] are, respectively, the mass, damping, and stiffness matrices of the dam and foundation. {ü

_{r}(t)}, $\left\{{\dot{u}}_{r}\left(t\right)\right\},$ and {u

_{r}(t)} are, respectively, the relative acceleration, velocity, and displacement. {ü

_{g}(t)} is the input earthquake acceleration from bedrock. [L

_{1}] is the conversion matrix between global coordinates and the local coordinates of the dam surface.

_{p}] is the additional mass matrix of hydrodynamic pressure. [L

_{2}] is the conversion matrix between global coordinates and the local coordinates of the river valley surrounding the reservoir.

_{p}] into the mass matrix of the dam [M

_{s}]. Then, a strong coupling method for a nonlinear cross-scale dam-reservoir system is established based on the polyhedron SBFEM.

## 4. Numerical Examples of Rigid Dams and River Valley

#### 4.1. Dams with Polygonal Mesh on Upstream Face

**a**, the density of water is expressed as

**ρ**, and the water depth is expressed as

**H**.

#### 4.2. Results and Discussion

## 5. Dynamic Coupling Analysis of Nonlinear Cross-Scale CFRD and Reservoir Systems

#### 5.1. Cross-Scale Model of the CFRD and Reservoir

#### 5.2. Material Parameters, Damping Methods, and Ground Motion

^{3}, elasticity modulus E = 25 GPa, Poisson’s ratios $\nu $ = 0.167) and bedrock (density ρ = 2.50 g/cm

^{3}, elasticity modulus E = 20 GPa, Poisson’s ratios $\nu $ = 0.2). An improved P–Z generalized plastic model was used for rockfills [59,67], of which the 17 material parameters were calibrated by the results of the triaxial tests and listed in the Table 1. Furthermore, an ideal elasto-plastic model was used to model the interface between the face slab and rockfills, of which the parameters are listed in Table 2. The compression stiffness of the slab and peripheral joints was 25,000 MPa/m, and the shear stiffness was 1 MPa/m. The Rayleigh damping method was employed for the various material and mechanical models of the CFRD.

^{2}. The results of the two conditions, which were considering the hydrodynamic pressure based on the polyhedron SBFEM condition and neglecting the hydrodynamic pressure condition, are compared and analyzed in the following section.

#### 5.3. Results and Discussion

_{p}] in Equation (34) was a null matrix, which meant that errors would occur because it was inconsistent with the actual situation. When calculating the error of the dynamic response of rockfill and concrete face slabs in the CFRD caused by ignoring the hydrodynamic pressure, the results under the condition of considering the hydrodynamic pressure were used as a standard. The compressive stress of the face slabs was positive.

#### 5.3.1. Rockfill

_{x}) and the dynamic displacement along the vertical direction (d

_{y}). As can be seen from Figure 21 and Figure 22 and Table 3, the distribution rules of the dynamic response for the rockfill were consistent under the two conditions of hydrodynamic pressure, but the differences of the maximum value and the corresponding size of the area with a large response were obvious. The dynamic acceleration and displacement of the rockfill were smaller when hydrodynamic pressure was considered.

#### 5.3.2. Concrete Face Slabs

## 6. Conclusions

- (1)
- A 3D hydrodynamic pressure calculation method based on the polyhedron SBFEM was proposed, in which the reservoir in front of a dam was simulated with polygonal semi-infinite prismatic fluid elements. The pre-processing of the reservoir model was simplified to a large extent, as the 3D mesh of the reservoir could be generated automatically from the 2D grid of the upstream face of dam. A high efficiency was achieved also by reducing the one-dimensional discretization. The proposed method has a high accuracy and provides a convenient numerical tool for a dynamic coupling analysis of a dam–reservoir system, when the cross-scale dam is modeled by the polyhedron SBFEM.
- (2)
- With an elastic–plastic CFRD being simulated by the polyhedron SBFEM and the hydrodynamic pressure of the reservoir being computed by the proposed polyhedron SBFEM for fluid, respectively, a nonlinear dynamic coupling method for cross-scale CFRD-reservoir systems based on the polyhedron SBFEM was developed. The results of a further numerical analysis showed that neglecting hydrodynamic pressure may produce obvious errors and lead to overestimation of the dynamic acceleration and displacement response of the rockfill, which is not conducive to an accurate and reasonable safety evaluation of a CFRD under an earthquake. Moreover, the hydrodynamic pressure had a big influence on the dynamic face slabs’ stresses, and the hydrodynamic pressure cannot be ignored in the dynamic stress analysis of face slabs.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

▽^{2} | Laplacian operator |

p | Hydrodynamic pressure |

ρ | Fluid density |

ü_{n} | Normal accelerations of the dam–reservoir interface |

ϋ_{n} | Normal accelerations of the river –valley interface |

$\left[N\left({\xi}_{2},{\xi}_{3}\right)\right]$ | Polygon mean-value shape function |

[J] | Jacobian matrix |

w | weight function |

[E^{0}], [E^{1}], [E^{2}], [C^{0}], [M^{1}] | Coefficient matrices |

$\left\{q\left({\xi}_{1}\right)\right\}$ | Nodal force |

[Z] | Hamilton coefficient matrix |

[Λ] | Eigenvalue matrix |

[Φ] | Eigenvector matrix |

[A] | The inverse of eigenvector matrix [Φ] |

${N}_{i}\left(x\right)$ | Interpolation function in mean-value coordinate system |

${w}_{i}\left(x\right)$ | Weight function of mean-value coordinate system |

$\Vert x-{x}_{i}\Vert $ | Eulerian distance between points |

W_{e} | Cartesian coordinate system |

W_{0} | Local coordinate system |

[M_{s}], [C_{s}], [K_{s}] | Mass, damping and stiffness matrices |

{ü_{r}(t)}, $\left\{{\dot{u}}_{r}\left(t\right)\right\}$, {u_{r}(t)} | Relative acceleration, velocity, and displacement |

{ü_{g}(t)} | Input earthquake acceleration |

[M_{p}] | Additional mass matrix of hydrodynamic pressure |

[L_{1}], [L_{2}] | Conversion matrix |

(x_{1}, x_{2}, x_{3}) | Global coordinates |

(ξ_{1}, ξ_{2}, ξ_{3}) | Local scaled boundary coordinates |

E | Elasticity modulus |

$\nu $ | Poisson’s ratios |

SBFEM | Scaled boundary finite element method |

CFRD | Concrete faced rockfill dam |

3D | Three-dimensional |

2D | Two-dimensional |

DOF | Degrees of freedom |

FEM | Finite element method |

BEM | Boundary element method |

PSBFEM | Polyhedron SBFEM |

PGA | Peak ground acceleration |

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**Figure 13.**Triangular valley with upstream–downstream excitation: (

**a**) along line AB and (

**b**) along line AC.

**Figure 16.**Semi-circular valley with upstream–downstream excitation: (

**a**) along line AB and (

**b**) along line BC.

**Figure 21.**Maximum distribution of rockfill acceleration along the upstream–downstream direction (m/s

^{2}): (

**a**) Neglecting hydrodynamic pressure and (

**b**) the Polyhedron SBFEM.

**Figure 22.**Maximum distribution of rockfill displacement along the vertical direction (m): (

**a**) Neglecting hydrodynamic pressure and (

**b**) the Polyhedron SBFEM.

**Figure 23.**Maximum distribution of concrete face slabs’ tensile stresses along the slope direction (MPa): (

**a**) Neglecting hydrodynamic pressure and (

**b**) the Polyhedron SBFEM.

**Figure 24.**Maximum distribution of concrete face slabs’ compressive stresses along the dam axial direction (MPa): (

**a**) Neglecting hydrodynamic pressure and (

**b**) the Polyhedron SBFEM.

G_{0} | K_{0} | M_{g} | M_{f} | α_{f} | α_{g} | H_{0} | H_{U}_{0} | m_{s} |

2400 | 2500 | 1.75 | 1.5 | 0.45 | 0.45 | 2900 | 2900 | 0.2 |

m_{v} | m_{t} | m_{u} | r_{d} | γ_{DM} | γ_{U} | β_{0} | β_{1} | |

0.28 | 0.2 | 0.25 | 105 | 70 | 7 | 50 | 0.023 |

k_{1} | k_{2} | n | φ/° | c/Pa |

300 | 1 × 10^{10} | 0.8 | 41.5 | 0 |

Hydrodynamic Pressure | Acceleration (m/s^{2}) | Displacement (m) | ||
---|---|---|---|---|

a_{x} | a_{y} | d_{x} | d_{y} | |

Polyhedron SBFEM | 4.61 | 2.72 | 0.062 | 0.055 |

Neglecting | 5.32 | 2.87 | 0.071 | 0.059 |

Error | 15.4% | 5.5% | 12.7% | 7.3% |

Hydrodynamic Pressure | Slope Direction (MPa) | Dam Axial Direction (MPa) | ||
---|---|---|---|---|

Tensile | Compressive | Tensile | Compressive | |

Polyhedron SBFEM | −3.83 | 3.16 | −2.05 | 3.38 |

Neglecting | −4.69 | 3.48 | −1.88 | 2.54 |

Error | 22.5% | 9.2% | 8.3% | 24.9% |

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**MDPI and ACS Style**

Xu, J.; Xu, H.; Yan, D.; Chen, K.; Zou, D.
A Novel Calculation Method of Hydrodynamic Pressure Based on Polyhedron SBFEM and Its Application in Nonlinear Cross-Scale CFRD-Reservoir Systems. *Water* **2022**, *14*, 867.
https://doi.org/10.3390/w14060867

**AMA Style**

Xu J, Xu H, Yan D, Chen K, Zou D.
A Novel Calculation Method of Hydrodynamic Pressure Based on Polyhedron SBFEM and Its Application in Nonlinear Cross-Scale CFRD-Reservoir Systems. *Water*. 2022; 14(6):867.
https://doi.org/10.3390/w14060867

**Chicago/Turabian Style**

Xu, Jianjun, He Xu, Dongming Yan, Kai Chen, and Degao Zou.
2022. "A Novel Calculation Method of Hydrodynamic Pressure Based on Polyhedron SBFEM and Its Application in Nonlinear Cross-Scale CFRD-Reservoir Systems" *Water* 14, no. 6: 867.
https://doi.org/10.3390/w14060867