Real-time surface manipulation with $$C^{1}$$ continuity through simple and efficient physics-based deformations

We present a novel but simple physics-based method to interactively manipulate surface shapes of 3D models with $$ C^1 $$ C 1 continuity in real time. A fourth-order partial differential equation involving a sculpting force originating from elastic bending of thin plates is proposed to define physics-based deformations and achieve $$ C^1 $$ C 1 continuity at the boundary of deformation regions. In order to obtain real-time physics-based surface manipulation, we construct a mapping relationship between a deformation region in a 3D coordinate space and a unit circle on a 2D parametric plane, formulate corresponding $$ C^1 $$ C 1 continuous boundary conditions for the unit circle, and obtain a simple analytical solution to describe the physics-based deformation in the unit circle caused by a sculpting force. After that, the obtained physics-based deformation is mapped back to the 3D coordinate space, and added to the original surface to create a new surface shape with $$ C^1 $$ C 1 continuity at the boundary of the deformation region. We also develop an interactive user interface as a plug-in of the 3D modelling software package Maya to achieve real-time surface manipulation. The effectiveness, easiness, real-time performance, and better realism of our proposed method is demonstrated by testing surface deformations on several 3D models and comparing with other methods and ground-truth deformations.

Diversity and Regularity of Periodic Impact Motions of a Mechanical Vibration System with Multiple Rigid Stops

The mechanical model of a two-degree-of-freedom vibration system with multiple rigid stops was established, and the effects of the multiple rigid stops to dynamic characteristics of two mass blocks of the system were studied. The judgment conditions and differential equations of motion of the system masses impacting rigid stops were analyzed. Based on the multiparameter and multiobjective collaborative simulation analysis, the correlation between the dynamic characteristics of the vibration system and the model parameters is studied. The basic periodic and subharmonic impact motions are analyzed with emphasis on the influences of dynamical parameters on the mode diversity and the distribution characteristics, and the law of emergence and competition of various periodic impact motions on the parametric plane is revealed. The singular points, the hysteresis transition domains, and the accompanying codimension-two bifurcations, caused by the irreversibility of the transition between adjacent basic periodic impact motions in the low-frequency domain, are analyzed. The reasonable parameter matching range, associated with dynamic characteristic optimization of the system, is determined.

Bifurcation Analysis of Composite Laminated Piezoelectric Rectangular Plate Structure in the Case of 1:2 Internal Resonance

In this paper, the authors study the bifurcation problems of the composite laminated piezoelectric rectangular plate structure with three bifurcation parameters by singularity theory in the case of 1:2 internal resonance. The sign function is employed to the universal unfolding of bifurcation equations in this system. The proposed approach can ensure the nondegenerate conditions of the universal unfolding of bifurcation equations in this system to be satisfied. The study presents that the proposed system with three bifurcation parameters is a high codimensional bifurcation problem with codimension 4, and 6 forms of universal unfolding are given. Numerical results show that the whole parametric plane can be divided into several persistent regions by the transition set, and the bifurcation diagrams in different persistent regions are obtained.

Interactive Flow-Induced Vibrations of Two Staggered, Low Mass-Ratio Cylinders in the TrSL3 Flow Regime (2.5 × 104 < Re < 1.2 × 105): Smooth Cylinders

Flow-induced vibrations (FIVs) of two elastically mounted circular cylinders in staggered arrangement were experimentally investigated. The Reynolds number range for all experiments (2.5 × 104 < Re < 1.2 × 105) was in the transition in shear layer 3 (TrSL3) flow regime. The oscillator parameters selected were: mass ratio m* = 1.343 (ratio of oscillating mass to displaced fluid mass), spring stiffness K = 250 N/m, and damping ratio ζ = 0.02. The experiments were conducted in the low turbulence free surface water (LTFSW) channel in the MRELab of the University of Michigan. A closed-loop, virtual spring–damper system (Vck) was used to facilitate quick and accurate parameter setting. Based on the characteristics of the displacement response, five vibration patterns were identified and their corresponding regions in the parametric plane of the in-flow spacing (1.57 < L/D < 4.57) and transverse cylinder spacing (0 < T/D < 2) were defined. The hydrodynamic forces and frequency characteristics of the vibration response are also discussed.

Interactive Flow-Induced Vibrations of Two Staggered, Low Mass-Ratio Cylinders in the TrSL3 Flow Regime (2.5×104

Flow-induced vibrations of two elastically mounted circular cylinders in staggered arrangement were experimentally investigated. The Reynolds number range for all experiments (2.5×104<Re<1.2×105) was in the TrSL3 flow regime. The oscillator parameters selected were: mass ratio m* = 1.343, spring stiffness K = 250N/m, and damping ratio ζ = 0.02. The experiments were conducted in the Low Turbulence Free Surface Water (LTFSW) Channel in the MRELab of the University of Michigan. A closed-loop, virtual spring-damper system (Vck) was used to facilitate quick and accurate parameter setting. Based on the characteristics of the displacement response, five vibration patterns were identified and their corresponding regions in the parametric plane of the in-flow spacing (1.57<L/D<4.57) and transverse cylinder-spacing (0<T/D<2) were defined. The hydrodynamic forces and frequency characteristics of the vibration response are discussed as well.

Characterization of uni-modal parametric plane curve singularities

In this article we characterize the classification of uni-modal parametric plane curve singularities given by Ishikawa and Janeczko, in terms of invariants. On the basis of this characterization we present an algorithm to classify the uni-modal parametric plane curve singularities and also give its implementation in computer algebra system SINGULAR.

King-Type Derivative-Free Iterative Families: Real and Memory Dynamics

A biparametric family of derivative-free optimal iterative methods of order four, for solving nonlinear equations, is presented. From the error equation of this class, different families of iterative schemes with memory can be designed increasing the order of convergence up to six. The real stability analysis of the biparametric family without memory is made on quadratic polynomials, finding areas in the parametric plane with good performance. Moreover, in order to study the real behavior of the parametric class with memory, we associate it with a discrete multidimensional dynamical system. By analyzing the fixed and critical points of its vectorial rational function, we can select those methods with best stability properties.

FittingC2Continuous Parametric Surfaces to Frontiers Delimiting Physiologic Structures

We present a technique to fitC2continuous parametric surfaces to scattered geometric data points forming frontiers delimiting physiologic structures in segmented images. Such mathematical representation is interesting because it facilitates a large number of operations in modeling. While the fitting ofC2continuous parametric curves to scattered geometric data points is quite trivial, the fitting ofC2continuous parametric surfaces is not. The difficulty comes from the fact that each scattered data point should be assigned a unique parametric coordinate, and the fit is quite sensitive to their distribution on the parametric plane. We present a new approach where a polygonal (quadrilateral or triangular) surface is extracted from the segmented image. This surface is subsequently projected onto a parametric plane in a manner to ensure a one-to-one mapping. The resulting polygonal mesh is then regularized for area and edge length. Finally, from this point, surface fitting is relatively trivial. The novelty of our approach lies in the regularization of the polygonal mesh. Process performance is assessed with the reconstruction of a geometric model of mouse heart ventricles from a computerized tomography scan. Our results show an excellent reproduction of the geometric data with surfaces that areC2continuous.