The isogeometric analysis method (IGA) is a new type of numerical method solving partial differential equations. Compared with the traditional finite element method, IGA based on geometric spline can keep the model consistency between geometry and analysis, and provide higher precision with less freedom. However, huge stiffness matrix from the subdivision progress still leads to the solution efficiency problems. This paper presents a multigrid method based on geometric multigrid (GMG) to solve the matrix system of IGA. This method extracts the required computational data for multigrid method from the IGA process, which also can be used to improve the traditional algebraic multigrid method (AGM). Based on this, a full multigrid method (FMG) based on GMG is proposed. In order to verify the validity and reliability of these methods, this paper did some test on Poisson’s equation and Reynolds’ equation and compared the methods on different subdivision methods, different grid degrees of freedom, different cyclic structure degrees, and studied the convergence rate under different subdivision strategies. The results show that the proposed method is superior to the conventional algebraic multigrid method, and for the standard relaxed V-cycle iteration, the method still has a convergence speed independent of the grid size at the same degrees.

In recent years, as a numerical method for solving partial differential equations (PDEs), isogeometric analysis [

The multigrid method [

It seems natural to extend these methods to IGA, and several remarkable works has been made in recent years. MG for IGA based on classical concepts have been considered in [

Based on previous generations, this paper uses the geometric information discretized by isogeometric subdivision to couple the MG with IGA and its application. Especially for the numerical algorithms such as IGA, we use geometric method to optimize them in the shortcomings of AMG, and it has integrated into the process of FMG.

Like FEM, IGA also have shape functions, not the Lagrange interpolation functions but the spline basis functions. As the two common spline functions used in CAD, B-spline and NURBS are briefly introduced below. With a given node vector, the B-spline basis function [

where the one-dimensional knot vector

If the basis function of B-spline is multiplied by an appropriate weight

FEA and IGA have the same theoretical basis (i.e., the weak form of partial differential equation). Take the common elliptic partial differential equation as an example:

where

It is assumed that the physical domain _{ij}

The weak form of the differential equation is obtained by multiplying with the trial function and integrating in the domain

The detailed process can be seen in [

Based on the requirements of light weight, modern CAD systems follow the principle of sufficient enough when characterizing models [

There are mainly three subdivision methods [

Taken the one-dimensional

where

If

where, the coefficients can be calculated as follows:

During

where

For FEM, most of MG method uses the algebraic multigrid method to construct interpolation operator and coarse the initial grid to get different hierarchical grid. In IGA, the geometric model is constructed from coarse to fine. There are different levels of grid naturally, and the mapping matrix between each level can be obtained from the subdivision process. In addition, each level has its own stiffness matrix and righthand vector. If this advantage is used to combine the multigrid method with IGA, the efficiency of solution would be improved.

In AMG, the main work is focused on constructing the interpolation matrix by coarsen the fine mesh, coarsening refined net _{m}_{m}

For MG, the mapping matrix is important. And its generation can affect the computational efficiency of MG directly. So, it is necessary to compare the generation rate of mapping matrix under different degrees of freedom. the time-consuming of mapping matrix generation (using Chang’s method) in

n | 100 | 324 | 1156 | 4356 | 16900 |
---|---|---|---|---|---|

Algebraic multigrid method |
0.169837 | 0.888387 | 4.669562 | 37.063700 | 569.107235 |

As discussed in Chapter 2.3, in the subdivision of IGA can generate the mapping relationship between new and old control points _{m}_{n}

in the formula,

The corresponding linear relation is that the non-zero element in row vector

Therefore, the extended matrix of

In theory, the restriction matrix ^{T}. But in fact, the mapping matrix

in the formula,

Since the control point

In IGA, the mapping matrix obtained by different refinement strategies are shown in

Continuation | ^{h} |
^{p} |
^{k} |

Restrictions | ^{h} |
^{p} |
^{k} |

Amended restrictions |

Based on one-dimensional subdivision, two-dimensional subdivision only needs to subdivide in

A time-consuming compare of mapping matrix generation between this method and AMG is given in

n | 100 | 324 | 1156 | 4356 | 16900 |
---|---|---|---|---|---|

Algebraic multiple method |
0.169837 | 0.888387 | 4.669562 | 37.063700 | 569.107235 |

Mapping matrix |
0.003802 | 0.009883 | 0.039448 | 0.199938 | 1.603531 |

The multigrid method is divided into initial period and calculation stage. In the initial period, GMG is based on the subdivision of the solution domain to get the grid of different scales. AMG only uses the information of coefficient matrix in algebraic equations to construct multigrid, and the method is consistent in the iterative solution stage. In order to improve the implement of AMG in IGA, geometric information will be introduced to AMG, what is called Geometric modification of algebraic multigrid method (GMG).

1. Initial period

The nested grid layer

Multigrid

2. Calculation stage

The iterative process of multigrid method (GMG) is given above. For a given parameter space ^{m} of each geometric level can be calculated directly. That means, GMG in this paper introduces geometric information as a more selection of residual and selects the smaller residual as the next relaxation iteration by comparing the residual between geometric grid and algebra grid.

As for the iteration of relaxation operator _{1} = _{2} = 2 represents the number of times of relaxation. In order to get the iteration structure, ^{m} is divided into ^{m} = ^{m} − ^{m}, and its iterative format can be written as:

^{m} is selected to obtain various iterative methods for relaxation. Jacobi, Gauss–Seidel and SOR iterations are commonly used. For rough grid correction, both the residual ^{m}^{m} = ^{m}, so ^{m −1} can be solved by the same relaxation operator

As known to all, ^{m} and ^{m} are discretized on each level grid in IGA, in order to make full use of these information, a full multigrid method (FMG) method suitable for IGA is established.

In the loop process of FMG, both AMG and GMG can be selected as solving algorithm on each layer, hence the accuracy of each layer can be guaranteed. Besides, and with the help of the extension matrix

In order to verify the validity of the algorithm, this paper has carried out the verification on the Poisson equation and Reynolds equation.

According to the relationship between residual norm and error norm:

The relative error can be expressed as follows:

The convergence rate is defined as:

The algorithm is written in OCTAVE, the computer processor is Core i5-4570, with 3.20 GHz frequency. We contrast the convergence and calculation time of different subdivision strategies. The unified cycle format is V cycle, the number of grid layers m = 3, and the subdivision mode of coarse grid to fine grid is _{1} = _{2} = 2.

The piston-cylinder liner is used as a lubrication system, and the oil film pressure is derived based on the Reynolds equation.

where _{0} and _{h}_{0} and _{h}

Since the lubrication area is symmetrical relative to the plane

Parameter/unit | Explanation | Value |
---|---|---|

Piston skirt radius | ||

Cylinder radius | ||

_{SK} |
Piston skirt length | |

Lubricant viscosity | ||

_{t} |
Eccentricity of upper end of piston skirt | |

_{b} |
Eccentricity of lower end of piston skirt | |

Derivative of _{t} |
||

Derivative of _{b} |
||

Reciprocating speed of piston | −10.6543 | |

_{b} |
Vertical distance from piston pin to piston crown | |

_{c} |
Distance from piston pin to piston centerline | 0 |

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | |
---|---|---|---|---|---|---|---|---|

0 | 4 | 9 | 25 | 81 | 289 | 1089 | 4225 | 16641 |

1 | 9 | 16 | 36 | 100 | 324 | 1156 | 4356 | 16900 |

2 | 16 | 25 | 49 | 121 | 361 | 1225 | 4489 | 17161 |

3 | 25 | 36 | 81 | 144 | 400 | 1296 | 4624 | 17424 |

The motion parameters of the piston-cylinder liner model have been given above, and the relevant detailed derivation process is referred to [

When the Reynolds equation is discretized, the knot vector of the initial parameter domain is

In the case of

n | _{2}-error |
Ration | n | _{2}-error |
Ration | n | _{2}-error |
Ration | n | _{2}-error |
Ration |
---|---|---|---|---|---|---|---|---|---|---|---|

81 | 5.7682e^{−9} |
0.0893 | 100 | 1.4610e^{−11} |
0.0420 | 121 | 5.4724e^{−10} |
0.4866 | 144 | 2.7144e^{−09} |
0.8618 |

289 | 2.6945e^{−10} |
0.0595 | 324 | 1.2098e^{−11} |
0.0438 | 361 | 9.3986e^{−11} |
0.4863 | 400 | 1.8839e^{−10} |
0.8599 |

1089 | 1.6173e^{−10} |
0.0553 | 1156 | 1.1407e^{−11} |
0.0467 | 1225 | 2.3928e^{−11} |
0.4867 | 1296 | 4.0223e^{−11} |
0.8405 |

4225 | 6.3934e^{−11} |
0.0536 | 4356 | 4.6670e^{−12} |
0.0453 | 4489 | 1.0456e^{−11} |
0.4835 | 4624 | 1.8401e^{−11} |
0.8400 |

16641 | 2.9877e^{−11} |
0.0534 | 16900 | 1.9957e^{−12} |
0.0444 | 17161 | 4.9974e^{−12} |
0.4830 | 17424 | 9.0104e^{−12} |
0.8402 |

n | Computation time/s | n | Computation time/s | ||
---|---|---|---|---|---|

AMG | GMG | AMG | GMG | ||

1089 | 0.01429 | 0.01381 | 1156 | 0.02011 | 0.02132 |

4225 | 0.04896 | 0.04004 | 4356 | 0.09043 | 0.08749 |

16641 | 0.19569 | 0.19189 | 16900 | 0.47305 | 0.45915 |

1225 | 0.06512 | 0.07268 | 1296 | 0.54113 | 0.56633 |

4489 | 0.35961 | 0.35194 | 4624 | 3.43262 | 3.32487 |

17161 | 2.00979 | 1.89226 | 17424 | 14.15859 | 13.95189 |

The equation of the two-dimensional Poisson problem with Dirichlet boundary conditions in the domain

The domain _{i}_{e}

For the parameter domain knot vector of two-dimensional Poisson equation is

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | |
---|---|---|---|---|---|---|---|---|

0 | 9 | 16 | 36 | 100 | 324 | 1156 | 4356 | 16900 |

1 | 16 | 25 | 49 | 121 | 361 | 1225 | 4489 | 17161 |

2 | 25 | 36 | 81 | 144 | 400 | 1296 | 4624 | 17424 |

n | _{2}-error |
Ration | n | _{2}-error |
Ration | n | _{2}-error |
Ration |
---|---|---|---|---|---|---|---|---|

100 | 3.6649e^{−11} |
0.053717 | 121 | 3.7055e^{−11} |
0.4703 | 144 | 4.3997e^{−11} |
0.8503 |

324 | 5.1800e^{−10} |
0.112660 | 361 | 1.5032e^{−12} |
0.4413 | 400 | 8.2384e^{−12} |
0.8505 |

1156 | 3.5185e^{−10} |
0.131564 | 1225 | 5.1833e^{−13} |
0.4587 | 1296 | 1.8076e^{−12} |
0.8351 |

4356 | 2.0245e^{−10} |
0.119191 | 4489 | 1.1490e^{−13} |
0.4633 | 4624 | 4.8159e^{−13} |
0.8362 |

16900 | 1.5931e^{−10} |
0.121456 | 17161 | 5.3937e^{−14} |
0.7366 | 17424 | 1.2852e^{−13} |
0.8485 |

n | Computation time/s | n | Computation time/s | n | Computation time/s | |||
---|---|---|---|---|---|---|---|---|

AMG | GMG | AMG | GMG | AMG | GMG | |||

1156 | 0.03075 | 0.02785 | 1225 | 0.06954 | 0.07617 | 1296 | 0.50252 | 0.52221 |

4356 | 0.10854 | 0.10078 | 4489 | 0.40158 | 0.37563 | 4624 | 3.31978 | 3.03466 |

16900 | 0.63829 | 0.58589 | 17161 | 2.18113 | 2.08475 | 17424 | 13.98436 | 12.82764 |

In this paper, the algebraic multigrid method and the geometric analysis of the natural discrete stiffness matrix

Based on the framework of IGA, the analytical models of Reynolds equation and Poisson equation are established, and multigrid is applied to the IGA. According to its solving process and method, a calculation program was developed in OCTAVE (FMG based on GMG modified the traditional AMG). The convergence rate of the iteration is independent of the size of the discrete mesh, so the multigrid method has the optimal computational complexity. When the degree of freedom of fine grid is determined, the degree of freedom of coarse grid can be reduced and the boundary information can be quickly spread to the whole, which can not only accelerate the convergence rate but also save the cost of calculation time. However, the selection of grid level also affects the convergence accuracy. In particular, the selection of the middle level has a greater impact on the results, and the performance of the middle grid close to the top grid is better. As for the combination of FMG and IGA, the results show that the algorithm can make full use of IGA to discretize the geometric information of each layer. Compared with the V-cycle, it has a faster initial convergence speed, but it is consistent with conventional algorithms in the later stage. How to make better use of the existing information to accelerate calculation efficiency and improve calculation accuracy is the direction we will continue to study in the future.