Data-driven optimization approach for mass-spring models parametrization based on isogeometric analysis

Highlights

• We present a NURBS Mass-Spring Model (NMSM) for simulating deformable objects.

• The Isogeometric Analysis (IGA) is applied to the NMSM parameterization.

• A sistematic procedure is developed to set up the NMSM parameters based on IGA.

• The parameters are determined so that the NMSM mimics a finite element model.

• We combine the simplicity and computational efficiency of the MSM with NURBS.

• We achieve a realistic deformation like as the IGA with reduced computational cost.

• Our proposal performs better than their counterparts based on classical FEM.

Abstract

The development of a systematic procedure to set up the parameters in a Mass-Spring Model (MSM) remains an open problem because the model parameters are not related to the constitutive laws of elastic material in an obvious way. One possibility to address this problem is to calculate MSM parameters from a reference model based on continuum mechanics and finite element (FEM) techniques. The traditional approaches in this area use isoparametric FEM, with linear shape functions, as the reference model. Recently, Isogeometric Analysis (IGA) has been used as new method for the analysis of problems governed by partial differential equations where Non-uniform Rational B-Splines (NURBS) are considered as basis of the analysis. Therefore, in this paper we propose a new method to derive MSM parameters using a data-driven strategy based on IGA approach. In this way, we propose a methodology for MSM derivation that is not restricted to a particular mesh topology and can consider higher order polynomial interpolation functions using the NURBS machinery. We validate the methodology for deriving MSM systems to simulate 2D/3D deformable objects. The obtained results are compared with related works in order to show the efficiency of our technique. We also discuss its robustness and issues against different NURBS geometry, order elevation, different discretizations and material properties.

Introduction

The mechanical behavior of elastic objects can be described by continuum elasticity theories that model how the objects deform under applied forces [1]. In the case of computer graphics applications, the search for models that enable real-time dynamic simulation of deformable objects is an active area of research [2], [3], [4], [5], [6], [7]. To perform computational simulation of the deformable body, the 3D/2D object's geometry is represented in computer aided design (CAD) files that must be translated into geometries suitable to be meshed in order to generate a compatible description for finite element (FEM) analysis [8]. Hence, as pointed out in [9], the geometric modeling and FEM communities realized the need to unify CAD and FEM descriptions which nowadays is known as Isogeometric Analysis (IGA) [10]. The idea of IGA analysis is that the functions used for the geometry description in CAD are adopted by the solution field representation (isoparametric concept).

In the specific case of non-uniform Rational B-Splines (NURBS) based IGA, the geometric description and field variables are represented via the same NURBS basis functions. Therefore, the control points associated with the NURBS basis function define the geometry and the degrees-of-freedom of field variables are the coefficients of the basis functions, i.e. control variables. The NURBS-based IGA framework [10] and extensions have been applied for vibrations, wave propagation, fluids, fluid–structure interaction, heat conduction, and elasticity problems [11], [12].

Other possibility for elastic objects simulation is to apply discrete approaches, based on Mass Spring Models (MSM), that represent the object's geometry by a mesh whose nodes are treated like mass points and edges act like springs connecting adjacent nodes. MSM approaches are preferable for real time applications because they are in general faster than the continuous ones besides being simple to implement [13]. Therefore, MSM techniques have been used to model deformable objects [2], for woven cloth simulation [13], [14], soft organic tissues, like muscles, face and abdomen in virtual surgery applications [15], [16], [17], [18], [19], [20].

Notwithstanding of the MSM advantages, it is known that methods that are based on the continuum mechanics and FEM are, in general, more realistic than their MSM counterparts [21], [22], [23]. That is the motivation to develop a general physically based or systematic method to determine the mesh topology or MSM parameters from the continuum set up. However, to the best of our knowledge, works in MSM derivation from FEM models have been restricted to linear interpolation functions [24], [25], [26]. On the other hand, NURBS based IGA enables accuracy of geometry representation and deals with higher-order polynomial basis functions [9]. These features can be exploited to improve the quality of MSM derivation processes.

Hence, in this paper, we propose a new derivation method for setting NMSM parameters (mass, spring stiffness coefficient and damping constants) using a data-driven strategy with a NURBS-based IGA model acting as continuum counterpart model. Due to this new approach, the mass spring model inherits NURBS parameterization in which the control points are treated like mass points linked by springs. For this reason, we have called our new scheme as NURBS Mass-Spring Model, abbreviated to NMSM. Generally speaking, this method computes the spring coefficients by solving an optimization problem based on the static equilibrium of NURBS-based IGA model and NMSM; calculates particles mass following [27], [28] and derives the damping constant using a technique based on [29].

The framework proposed in this work determines the NMSM parameters so that it behaves like IGA (its continuum counterpart), eliminating the trial-and-error approach for the parameters tuning. The NMSM adds relevant contributions in the state-of-the art in MSM parameterization approaches: (a) It allows to increase the geometric accuracy in the representation of elastic objects, since we have a mass-spring model based on NURBS; (b) it can simulate deformations of curved shapes by exploring NURBS refinement techniques or manipulation of higher order polynomials; (c) the method combines the simplicity and computational efficiency of the MSM with NURBS to achieve deformation simulation with realism similarly to the one offered by NURBS-based IGA model but with reduced computational cost.

We validate the proposed method by several experiments in the context of 2D and 3D IGA models. The obtained results show that the derivation methodology has the lowest percentage of error when compared with related ones. Also, we verify the robustness of the algorithm against different discretizations and NURBS geometry, optimization problem setups and material properties. Respect to the polynomial order, the method achieves outstanding results for first and second orders. However, we notice a degradation in the ability of the derived NMSM to mimic the IGA model when increasing the NURBS polynomial order.

The paper is organized as follows. Section 2 describes the related literature. The NMSM approach is presented in Section 3. Experimental results are presented in Section 4. Next, Section 5 discusses the experimental results and related features of NMSM. Then, in Section 6, the conclusions and future works are presented. Appendix A offer some details about the implementation, which was carried out in MATLAB, software architecture, and parameters/options used in the simulated annealing function. Finally, Appendix B presents some background in NURBS based IGA models.