A novel deformable B-spline curve model based on elasticity

Abstract

The physically based deformable curve models are widely used to simulate thin one-dimensional objects in computer graphics, interactive simulation, and surgery simulation. These models consider objects to be rods described by an adapted frame curve that contains the rod’s centerline as well as the orthonormal material frame of each point on the centerline. However, they pose challenges including fine discretization, redundancy in modeling slender rods, and maintaining accuracy and stability. In this paper, we propose a novel physically based deformable B-spline curve model that regards curves as rods consisting of parallel fibers and derive elastic potential energy only from curves’ representations. Therefore, our model does not take rotation-based adapted frames into consideration and reduces degree of freedom. Our model divides the curves into infinitesimal elements in parameter space and derives the analytical relationship between elastic potential energy function and curves’ representations through the change of total length of infinitesimal elements’ fibers. Our model can support material attributes in the real world and maintain the reality and stability of the solution. We employ isogeometric analysis to solve the dynamic equations derived from our deformable model as isogeometric analysis is suitable to solve the dynamic equations of parametric models. We compare the scenarios in the real world, our model’s simulation results, and other model’s results to demonstrate the reality of our models. The results are in line with expectation. We design several examples to demonstrate our models’ applications.

Appendices

Appendix A: The derivative of external potential energy

Suppose the surface is under conservative forces, so the potential energy has general functions:

$$\begin{aligned} E_{\text{ ext }}=\int {{\textbf {F}}}(x,y,z)du \end{aligned}$$

(A1)

where F(x, y, z) is the potential density function. We derive the expression of the derivative through the Jacobian matrix. For arbitrary generalized coordinates $p_{k}$, the derivative $\frac{\partial E_{\text{ ext }}}{\partial p_{k}}$ can be transformed by the chain rule:

$$\begin{aligned} \begin{aligned} \frac{\partial E_{\text{ ext } }}{\partial p_{k}}&=\int \left[ \frac{\partial x}{\partial p_{k}}, \frac{\partial y}{\partial p_{k}}, \frac{\partial z}{\partial p_{k}}\right] \left[ \frac{\partial {{\textbf {F}}}}{\partial x}, \frac{\partial {{\textbf {F}}}}{\partial y}, \frac{\partial {{\textbf {F}}}}{\partial z}\right] ^{T} \end{aligned} \end{aligned}$$

(A2)

By integrating the expression of $\frac{\partial E_{\text{ ext }}}{\partial p_{k}} (k=0,1,...n)$, we can derive $\frac{\partial E_{\text{ ext }}}{\partial P}$ as follows:

$$\begin{aligned} \frac{\partial E_{\text{ exp } }}{\partial {{\textbf {P}}}}{} & {} =\int {{\textbf {J}}}^{T}\left( \frac{\partial {{\textbf {F}}}}{\partial x},\frac{\partial {{\textbf {F}}}}{\partial y},\frac{\partial {{\textbf {F}}}}{\partial z}\right) ^{T} du \nonumber \\{} & {} =-\int {{\textbf {J}}}^{T}{{\textbf {f}}}(x,y,z)du \end{aligned}$$

(A3)

where f(x, y, z) is the external force field density.

Appendix B: Gauss–Legendre quadrature

Gauss–Legendre quadrature is a method that provides a discrete approximation of the definite integral of a function. For integrating over the interval $[-1,1]$, the rule is stated as:

$$\begin{aligned} \int _{-1}^{1} f(x) d x \approx \sum _{i=1}^{n} \omega _{i} \textrm{f}\left( x_{i}\right) \end{aligned}$$

(B4)

The parameter n represents the number of sample points and $\omega _{i}$ are quadrature weights. $x_{i}$ are the roots of the nth Legendre polynomial.

We employ Gauss–Legendre quadrature to obtain the quadrature of the function on the right side of the equation and can further derive the quadrature of the original function from the above equation. To balance accuracy and computational time, we choose three sample points and the form of the quadrature formula is shown as follows:

$$\begin{aligned} \int _{-1}^{1} f(x) d x \approx \frac{5}{9} f\left( -\frac{\sqrt{15}}{5}\right) +\frac{8}{9} f(0)+\frac{5}{9} f\left( \frac{\sqrt{15}}{5}\right) \nonumber \\ \end{aligned}$$

(B5)

We define $\omega _{1} = \omega _{3} = \frac{5}{9}$, $\omega _{2} = \frac{8}{9}$. For parameter u whose domain is [0, 1], we define that $u_{1} = (1-\frac{\sqrt{15}}{5}) / 2$, $u_{2} = 0.5$, $u_{3} = (1 + \frac{\sqrt{15}}{5}) / 2$.

For B-spline curve, the form of the quadrature formula is:

$$\begin{aligned} \int _{0}^{1} f(u) du \approx \sum _{i=1}^{3} \omega _{i} \textrm{f}\left( u_{i}\right) \end{aligned}$$

(B6)

where the values of $\omega _{i}$ and $u_{i}$ are listed above.