Geometrically exact dynamic splines
Introduction
One-dimensional flexible models are a key CAD element in number of practical situations. Cables of largely varying mechanical properties are nowadays used in industry. In fields such as car and plane design, virtual prototyping is used to improve quality and to reduce development costs. As a matter of fact, virtual prototyping includes more and more assembly simulations: it allows the early detection of potential problems, and also permits the study of ease of assembly. This implies the ability to accurately represent geometry, but also the mechanical behavior of involved parts. Among the many objects to be simulated, flexible one-dimensional objects are of significant importance. They are involved in vehicle engineering (e.g. electrical cable laying within the car structure [1]), but also in fields such as architecture (e.g. stiff electrical cable positioning within virtual buildings), or even in medical simulation (surgical thread simulation is currently an active research question in the medical simulation community [2]). For most unconstrained Computer Aided Design applications, splines are probably the most classical tool for 1D objects. As a matter of fact, NURBS have become an industry-standard representation for 1D objects. Dynamic splines have been introduced by Qin and Terzopoulos [3]. They combine physics-based constraining equations with spline geometry, in order to improve the design process. In this article, we propose an approach that extends the mechanical accuracy of previously proposed approaches. We propose, wherever possible, geometrically exact formal expressions that, along with spline analytical expressions, to provide a powerful, real-time model. We call this model Geometrically Exact Dynamic Splines, or GEDS for short.
In this paper, we propose a spline-based model for the real-time, mechanically accurate, simulation of one-dimensional objects. Our model can handle both reversible (elastic) and irreversible (plastic) deformations. The proposed formalism and energy expressions model stretching, bending and twisting loads; at the very limit of material constraint, we show how our model can be used to detect break points. We also show that the interactive rate is provided for a wide range of configurations. Finally, we describe a practical application of our model, oone that permits us to virtually validate electrical cable positioning and clipping along a path on a car door. Our specific scientific contributions are the following:
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An improvement in the dynamic spline state-of-the-art, namely a formulation of stretching, twisting and bending deformations in large rotations (or large displacements), through geometrically exact energy expressions. Such terms allow accurate results, while still running in interactive times. The proposed method is fully compatible with Lagrangian multipliers;
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The handle of twisting with an extended spline formulation, by decomposition of twisting in a geometrical part and a roll part. Such a separation ensures numerical stability for twisting energy evaluation. In addition to that, solving of the proposed mechanical model does not require a local frame, which makes it all the more accurate (frames are classically stabilized along a 1D curve using non-mechanical methods, see Section 8);
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An easy and efficient inclusion of plasticity within the Lagrange spline model.
The remainder of the paper is organized as follows: next section describes related work. In Section 3, we provide a short summary of elasticity and plasticity theory, which constitutes the core mechanical knowledge for understanding the remainder of the article. Then we define in Section 4 the formalism we use to describe geometrical model configurations. In Section 5, we propose a method (including elegant formalism) for handling elastic deformations of the model in a geometrically exact manner. In Section 6, we show how elastic deformation simulation can be combined with plastic behavior detection and simulation. For the sake of completeness, we provide in Section 7 the (classical) tools we use to handle the world’s interaction with the 1D model. Section 8 provides some comments about twisting handling in the proposed deformation model, which is one of the very crucial points in the method. Finally, Section 9 describes tests and practical results: first, for ease of understanding, we provide a complete overview of the animation algorithm, which links the equations all together. Second, we compare numerical results of our model to several classical reference configurations. Third, we describe an advanced practical application of the model: a virtual system for electrical cable position testing on a car door (see visual example of Fig. 1).
Section snippets
Previous works
A constraints solver has now become a standard part of most CAD models, and is still a very active research field [4]; the range of applications of such techniques is potentially very large (e.g see [5]). Constraint solving most often relates to finding a compatible solution between user modeling requirements and pre-imposed geometric constraints. Variational modeling[6] minimizes the global energy of a constrained geometric deformable object, and can be seen as an introduction of physical
Mechanics background
This section provides a very short overview of the mechanical background, which is necessary for understanding the extension of Dynamic spline we propose in the next section.
When a force is applied to a deformable object, object geometry is extended or compressed, and local topology may even change (i.e. material might break) if the force is large enough. Although both load and extension/compression are primary quantities, material scientists tend to use two derived quantities, stress and
Beam geometry definition
Beam theory is the study of one-dimensional objects in mechanics. Consider a cross-section of diameter and area , as shown in Fig. 3. The neutral fiber or neutral axis, denoted , is the oriented curve of length that passes through the center of every cross-section. The volume defined by these cross-sections is a beam.
Spline formulation
The beam configuration is entirely described by two fields: a position field , which determines the neutral fiber position, and a rotation field , which provides
GEDS in the elastic domain
To obtain the motion of control points with the Lagrange equations, deformation energies must be first formulated from physical parameters, and then differentiated with respect to the degrees of freedom. In this section, we propose a unified formulation to describe the deformations of a one dimensional object and the exact calculation of the corresponding forces.
GEDS in the plastic domain
In this paper, we may also treat one-dimensional objects as perfectly plastic and breakable, with a stress–strain curve of the form shown in Fig. 2 with perfect plasticity. It is possible to simulate real plasticity using a function of and , which gives the part of the force to convert into strain . In practice, we used ideal plasticity materials: as one can see Section 9.4, our practical results show convincing simulations using ideal plasticity. In addition to that, it is known that
World interaction
The spline-based model is continuous, that is, mechanically defined everywhere along the one-dimensional object. An applied force on the point of the spline provides generalized forces . Differentiating the power with respect to the generalized coordinate yields the corresponding generalized force : A force may consequently be applied everywhere, but interacting with the manipulated object remains quite difficult. This is the reason why we use Lagrangian
From mechanical point of view
A cross-sectional orientation field is not required to solve mechanics, but only to visualize twisting and apply textures. As a matter of fact, bending and geometrical twisting only depend on control point positions, whereas the roll is not directly considered in the mechanical equations, but its derivative with respect to the spline parameter . A major convenience of our model is that its accuracy does not rely on frames. This allows a real continuity of the one-dimensional object.
From visual point of view
However, we
Implementation
The overall algorithm 2 recalls the required steps to simulate a physically-based spline, including elasticity and plasticity.
A one-dimensional object is completely defined by a spline specified by an arbitrary number of control points as well as by some physical parameters. These parameters comprise the cross-section diameter and material density , as well as two parameters of elasticity (that is, two of the three interrelated constants of Young’s modulus, shear modulus and Poisson’s
Conclusion and future work
Using a background in mechanics consisting of elasticity and plasticity theories, we have proposed a deformable model for one-dimensional objects. Our approach addresses reversible and irreversible deformations, like stretching, twisting and bending, and can even detect fractures. This model provides both accurate mechanical simulation as well as quick calculations. Moreover, we can impose positions and orientations everywhere along the object. We can also simulate a wide range of materials in
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