Isogeometric analysis (IGA) is introduced to establish the direct link between computer-aided design and analysis. It is commonly implemented by Galerkin formulations (isogeometric Galerkin, IGA-G) through the use of nonuniform rational B-splines (NURBS) basis functions for geometric design and analysis. Another promising approach, isogeometric collocation (IGA-C), working directly with the strong form of the partial differential equation (PDE) over the physical domain defined by NURBS geometry, calculates the derivatives of the numerical solution at the chosen collocation points. In a typical IGA, the knot vector of the NURBS numerical solution is only determined by the physical domain. A new perspective on the IGA method is proposed in this study to improve the accuracy and convergence of the solution. Solving the PDE with IGA can be regarded as fitting the load function defined on the NURBS geometry (right-hand side) with derivatives of the NURBS numerical solution (left-hand side). Moreover, the design of the knot vector has a close relationship to the NURBS functions to be fitted in the area of data fitting in geometric design. Therefore, the detected feature points of the load function are integrated into the initial knot vector of the physical domain to construct the new knot vector of the numerical solution. Then, they are connected seamlessly with the IGA-C framework for its great potential combining the accuracy and smoothness merits with the computational efficiency, which we call

Isogeometric analysis (IGA) [

IGA has been initially implemented by Galerkin formulations (isogeometric Galerkin (IGA-G)) [

When PDEs are solved over a physical domain defined by a NURBS geometry, typical IGA commonly takes the NURBS as the bases of the approximation solution space, which is the same as the NURBS bases that construct the physical domain space. The NURBS bases are determined totally by the knot vector [

A new method of IGA is proposed in this study to improve the accuracy when PDEs are solved over the physical domain defined by NURBS geometry. We exploit the idea that the knot vector of the NURBS numerical solution is decided together by the load function and geometry representation. The feature points of the load function over the physical domain are detected in 1D, 2D, and 3D cases. Then, we merge these feature points into the initial knot vector of the spline representation of the physical domain. Given the great potential of the computational efficiency of IGA-C, we focus on the performance of the IGA-C framework with our generated knot vector. Thus, we name it as

The rest of this paper is structured as follows:

The standard IGA method is implemented by Galerkin formulation (IGA-G) based on the weak form of the problem. The stiffness matrix assembly involves many numerical integrations, such as the commonly used Gauss quadrature. The integration efficiency in IGA has been first considered by [

The research of IGA-C is mostly limited by the options of the ideal collocation points. In IGA-C, the commonly used collocation points are Greville abscissae [

In CAD, the physical domain is expressed by a NURBS mapping from a parametric domain

We name

In particular, let the following Laplace equation over the physical domain defined by a NURBS geometry

The IGA-G method ([

Galerkin’s method aims to construct the finite-dimensional approximations of the spaces

Then the weak form can be transformed into

By assembling the system of the weak form

The IGA-C method ([

IGA-C solves the equations by introducing a set of collocation points

When

The IGA-G and IGA-C methods involve the derivatives of the NURBS basis functions when the numerical solution

Therefore, the feature points on the load function

Given the great potential of computational efficiency of IGA-C, this study focuses on the performance of IGA-C framework with our generated knot vector. We propose the IGA-CL method, which combines the representation of the physical domain and the load function characteristics to construct the knot vector for the numerical solution in the IGA-C framework. The main idea is illustrated in

In this section, we introduce the detecting method of feature points from the load functions of the PDEs defined on a physical domain represented by NURBS geometry. We develop a method of knot generation to construct the knot vector once the feature points are detected.

We begin with the detecting method of the load function of the PDE in a 1D case. Comparing with uniform sampling methods using arc length and curvature information ([

First, the physical domain

The seed points include endpoints, inflection points, and extreme curvature points, which are determined by referring to [

Then

The seed point detection is concluded in Algorithm 1. As illustrated, the domain [0, 1] of the parametric curve of the load function

The seed point set

Define

Apparently,

The feature point detection is presented in Algorithm 2. Algorithm 1 returns

2D case is regarded as the extension of the 1D case along the

A load function

The mean curvature of the two principle curvatures is good for reflecting the surface shape. Nevertheless, we maintain the curvature computed in each direction without loss of generality. Fix

The curvature of the load function

Subsequently, we express the marginal cumulative integrals of curvature along the

Thus, the marginal cumulative characteristic functions along the

Based on

For the 3D case, we construct the characteristic functions in a way similar to that in the 2D case. The load function is

The volume of the parametric solid is presented as below:

Thus, the marginal cumulative volumes in three directions can be described as

On the other hand, the curvatures in

Then, the curvature of the load function

The marginal cumulative integrals of the curvature in

Therefore, the marginal cumulative characteristic functions of the load function

Given that the process is similar to surface detection, we omit the details here.

Many numerical approaches were proposed to solve PDEs, among which IGA-C is an efficient method, which is described in

Given the 1D scenario, the other two cases are treated similarly. Since the point detection is done, we obtain the feature point set

Moreover, Hosseini et al. summarized three knot placement techniques for geometric construction from the data points to decide the positions of the internal knots of the knot vector ([

The spline numerical solution is then expressed as

Employing collocation schemes such as Greville abscissae, superconvergent points, and Cauchy-Galerkin points yields a set of collocation points. Replacing the unknown function

In this section, we validate the influence of computing accuracy with our isogeometric analysis by fitting load functions in solving PDEs through numerical experiments. The typical IGA-C and our IGA-CL are compared in

Given the advantage of our IGA-CL method, we utilize the feature points of the load function and construct an integrated knot vector. To verify the accuracy improvement, we apply our IGA-CL framework to solve

Let the ordinate be the

Our method demonstrates its superiority over typical IGA-C in terms of accuracy under all the degrees

For PDEs under other boundary conditions, we present the following example:

In this case, we set the parameter

We compare IGA-CL with IGA-C at Greville abscissae and SC points in solving the following two 2D Examples III and IV. The main idea of the IGA-CL is detecting the feature points on the load function and then linking them to IGA-C for solving PDEs. Whether the accuracy of the IGA-CL solution outperforms IGA-C principally depends on the quality of the feature points detected. The weights in the characteristic functions can be controlled and adjusted, thereby influencing the detection effect. We show the convergence of examples with a relatively satisfying choice of weight.

The load function is

The load function is

To begin with, we present

In order to solve the elliptic equations Examples III and IV, the IGA-CL and IGA-C methods at Greville and SC collocation points are employed and compared in terms of absolute error. The initial number of detected points is

However, in these two cases, considering the extreme nonuniformity after h-refinements when a small weight is adopted, we take a large weight in practical experiments to ensure that the flat part of the shape of load function has fairly and evenly detected points. For simplicity, the weights we adopted here are

With the appropriate weights, we exhibit the convergence comparisons of the relative error of IGA-CL and IGA-C at Greville abscissae and SC points under different degrees of NURBS bases of the numerical solution of Examples III and IV in

In this case, the initial number of feature points that we sampled in our IGA-CL method is 6 for each direction, and the weights in characteristic functions are

Then, we compare the distribution of absolute errors between IGA-CL and IGA-C at Greville abscissae with the same

The convergence figures of relative errors are also shown.

One concern that should be addressed is that the detection of the feature points as a pre-processing costs time. Here we show the comparison of the time cost and relative error between IGA-CL and IGA-C in

Examples | DoFs | IGA-CL | IGA-C | |||
---|---|---|---|---|---|---|

Detection | Computation | Relative error | Computation | Relative error | ||

1D: Example I | 7684 | 20.099 | 21.992 | 9.5477e-11 | 22.603 | 1.8522e-10 |

1D: Example II | 8197 | 6.704 | 25.399 | 7.4817e-10 | 24.973 | 3.6400e-10 |

2D: Example III | 12100 | 35.856 | 150.594 | 1.1580e-7 | 151.332 | 3.7566e-7 |

2D: Example IV | 14161 | 36.466 | 293.799 | 3.7136e-7 | 292.682 | 9.2534e-6 |

3D: Example V | 12167 | 65.214 | 632.034 | 8.1771e-8 | 633.725 | 1.5276e-7 |

Based on the knot selection scheme, IGA concentrating on load function is applicable not only to the IGA-C method, but also to the IGA-G method. For Example I in

This study proposes a new perspective on solving IGA problems, which treats the left-hand side and right-hand side of the PDE as fitting the load function with the derivatives of the unknown NURBS-expressed solution. We present a detailed introduction to how to detect feature points on the load function of the PDE in 1D, 2D, and 3D cases. Then, we propose the IGA-CL method in which the knot vector of the NURBS numerical solution combines the characteristics of the load function with the representation of the physical domain. We validate that utilizing the feature points of the load function to construct the integrated knot vector helps improve the accuracy while generally maintaining the convergence rate. The theoretical result of the convergence behavior is beyond the scope of this study. Experimental results demonstrate that the accuracies of both our IGA-CL framework and IGA-GL framework are higher than those of the typical IGA-C and IGA-G methods, particularly in solving problems with analytical solutions with obvious characteristics.

This work is supported by the

The authors declare that they have no conflicts of interest to report regarding the present study.

^{−}and F

^{−}projection methods for nearly incompressible linear and non-linear elasticity and plasticity using higher-order NURBS elements