Designing and Mapping Trimming Curves on Surfaces Using Orthogonal Projection

1990 ◽  
Author(s):  
Joseph Pegna ◽  
Franz-Erich Wolter
Keyword(s):  
Differential Equation ◽  
Orthogonal Projection ◽  
Broad Class ◽  
Design Tool ◽  
Space Curve ◽  
Computationally Efficient ◽  
Curves On Surfaces ◽  
Surface Representations ◽  
Differential Properties ◽  
Surface Curve

Abstract In the design and manufacturing of shell structures it is frequently necessary to construct trimming curves on surfaces. The novel method introduced in this paper was formulated to be coordinate independent and computationally efficient for a very general class of surfaces. Generality of the formulation is attained by solving a tensorial differential equation that is formulated in terms of local differential properties of the surface. In the method proposed here, a space curve is mapped onto the surface by tracing a surface curve whose points are connected to the space curve via surface normals. This surface curve is called to be an orthogonal projection of the space curve onto the surface. Tracing of the orthogonal projection is achieved by solving the aforementionned tensorial differential equation. For an implicitely represented surface, the differential equation is solved in three-space. For a parametric surface the tensorial differential equation is solved in the parametric space associated with the surface representation. This method has been tested on a broad class of examples including polynomials, splines, transcendental parametric and implicit surface representations. Orthogonal projection of a curve onto a surface was also developed in the context of surface blending. The orthogonal projection of a curve onto two surfaces to be blended provides not only a trimming curve design tool, but it was also used to construct smooth natural maps between trimming curves on different surfaces. This provides a coordinate and representation independent tool for constructing blend surfaces.

Surface Curve Design by Orthogonal Projection of Space Curves Onto Free-Form Surfaces

1996 ◽  
Vol 118 (1) ◽  
pp. 45-52 ◽  
Author(s):  
Joseph Pegna ◽  
Franz-Erich Wolter
Keyword(s):  
Differential Equation ◽  
Broad Class ◽  
Free Form ◽  
Implicit Surface ◽  
Computationally Efficient ◽  
Surface Representations ◽  
Isoparametric Maps ◽  
Free Form Surfaces ◽  
Differential Properties ◽  
Surface Curve

A novel technique for designing curves on surfaces is presented. The design specifications for this technique derive from other works on curvature continuous surface fairing. Briefly stated, the technique must provide a computationally efficient method for the design of surface curves that is applicable to a very general class of surface formulations. It must also provide means to define a smooth natural map relating two or more surface curves. The resulting technique is formulated as a geometric construction that maps a space curve onto a surface curve. It is designed to be coordinate independent and provides isoparametric maps for multiple surface curves. Generality of the formulation is attained by solving a tensorial differential equation formulated in terms of local differential properties of the surfaces. For an implicit surface, the differential equation is solved in three-space. For a parametric surface the tensorial differential equation is solved in the parametric space associated with the surface representation. This technique has been tested on a broad class of examples including polynomials, splines, transcendental parametric and implicit surface representations.

Nearly Exact Bayesian Estimation of Non-linear No-Arbitrage Term-Structure Models*

2021 ◽  
Author(s):  
Marcello Pericoli ◽  
Marco Taboga
Keyword(s):  
Bayesian Estimation ◽  
Term Structure ◽  
Broad Class ◽  
Model Parameters ◽  
State Variables ◽  
Computationally Efficient ◽  
Macroeconomic Factors ◽  
No Arbitrage ◽  
Term Structure Models ◽  
Non Linear

Abstract We propose a general method for the Bayesian estimation of a very broad class of non-linear no-arbitrage term-structure models. The main innovation we introduce is a computationally efficient method, based on deep learning techniques, for approximating no-arbitrage model-implied bond yields to any desired degree of accuracy. Once the pricing function is approximated, the posterior distribution of model parameters and unobservable state variables can be estimated by standard Markov Chain Monte Carlo methods. As an illustrative example, we apply the proposed techniques to the estimation of a shadow-rate model with a time-varying lower bound and unspanned macroeconomic factors.

scholarly journals FPGA Based Engine Control Module for Fuel Injection System

2019 ◽  
Vol 8 (10) ◽  
pp. 3888-3892
Keyword(s):  
Fuel Injection ◽  
Design Tool ◽  
Operating Conditions ◽  
Injection System ◽  
Fuel Injection System ◽  
Engine Control ◽  
Computationally Efficient ◽  
Control Module ◽  
Fuel Prices ◽  
Ecm Architecture

Fuel injection system is an indispensible part of the present day automobiles. The depletion of the fuels along with continuous surge in the fuel prices has made it imperative to use fuel economically and restricting the wastage to a minimum. Contrary to the carburetor, using predefined amount of fuel irrespective of the environment, Fuel Injection System uses just the required amount of fuel based on the operating conditions as sensed by the Engine Control Module (ECM). Numerous parameters are required to be sensed by the ECM to achieve optimum efficiency of the engine. To handle the processing of such large number of parameters, a robust architecture is required. This paper presents the design and implementation of ECM utilized in Electronic Fuel Injection (EFI) system on a Field Programmable Gate Array. The ECM architecture discussed in the proposed system is computationally efficient enough to fulfill ever-increasing functionalities of the ECM. The main objective of this research is to sense the parameters required for the ECM analysis and to interpret and analyze this data and accordingly control the solenoid (actuator). The CAN controller is also deployed in an FPGA to facilitate the communication between ECM and Human Machine Interface (HMI) to indicate the parameters sensed by the sensor on the LCD. The target device (FPGA) for this work is Xilinx Spartan 3E and the design tool is Xilinx ISE 14.7 with the ECM and CAN controller being modeled in Verilog Hardware Description Language (HDL).

scholarly journals A Picard-Maclaurin theorem for initial value PDEs

2000 ◽  
Vol 5 (1) ◽  
pp. 47-63 ◽  
Author(s):  
G. Edgar Parker ◽  
James S. Sochacki
Keyword(s):  
Differential Equation ◽  
Series Solution ◽  
Initial Conditions ◽  
Broad Class ◽  
Picard Iteration ◽  
Power Series Solution ◽  
Initial Value ◽  
Arbitrary Degree ◽  
Cauchy Theorem ◽  
Analogous Theorem

In 1988, Parker and Sochacki announced a theorem which proved that the Picard iteration, properly modified, generates the Taylor series solution to any ordinary differential equation (ODE) onℜnwith a polynomial generator. In this paper, we present an analogous theorem for partial differential equations (PDEs) with polynomial generators and analytic initial conditions. Since the domain of a solution of a PDE is a subset ofℜn, we identify one component of the domain to achieve the analogy with ODEs. The generator for the PDE must be a polynomial and autonomous with respect to this component, and no partial derivative with respect to this component can appear in the domain of the generator. The initial conditions must be given in the designated component at zero and must be analytic in the nondesignated components. The power series solution of such a PDE, whose existence is guaranteed by the Cauchy theorem, can be generated to arbitrary degree by Picard iteration. As in the ODE case these conditions can be met, for a broad class of PDEs, through polynomial projections.

scholarly journals Slant Helices of (k,m)-Type According to the ED-Frame in Minkowski 4-Space

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2185
Author(s):  
Fatma Bulut
Keyword(s):  
Differential Geometry ◽  
Minkowski Space ◽  
Symmetric Matrix ◽  
Space Curve ◽  
Frame Field ◽  
Dimensional Minkowski Space ◽  
Darboux Frame ◽  
Surface Curve ◽  
Area Of Study

In differential geometry, relations between curves are a large and important area of study for many researchers. Frame areas are an important tool when studying curves, specially the Frenet–Serret frame along a space curve and the Darboux frame along a surface curve in differential geometry. In this paper, we obtain slant helices of k-type according to the extended Darboux frame (or, for brevity, ED-frame) field by using the ED-frame field of the first kind (or, for brevity, EDFFK), which is formed with an anti-symmetric matrix for ε1=ε2=ε3=ε4∈{−1,1} and the ED-frame field of the second kind (or, for brevity, EDFSK), which is formed with an anti-symmetric matrix for ε1=ε2=ε3=ε4∈{−1,1} in four-dimensional Minkowski space E14. In addition, we present some characterizations of slant helices and determine (k,m)-type slant helices for the EDFFK and EDFSK in Minkowski 4-space.

Surface-Curve Contact in 4-Axis NC Machining of Sculptured Surfaces Using Anural Cutter

2010 ◽  
Vol 37-38 ◽  
pp. 1372-1375
Author(s):  
Hu Ran Liu
Keyword(s):  
Differential Equation ◽  
Nc Machining ◽  
Sculptured Surfaces ◽  
End Mill ◽  
Ball End Mill ◽  
Axis Orientation ◽  
Milling Tool ◽  
Swept Surface ◽  
Complicated Surface ◽  
Surface Curve

This is the highly effective method. In this paper an innovative theory for machining complicated surface is presented. By using a anural milling tool instead of ball-end mill or the flat-ended tool, and by adjusting the axis of cutter relative to the surface, the two surfaces, the swept surface and the required surface, have the same curvatures, up to as high as 3rd order. Through the deduction of differential equation, some theory on partial touching between surfaces and surfaces when manufacturing has been explored. The problem of axis orientation under this condition has also been discussed clearly. The outside of the circular tool is a surface; the surface to be machined is a groove, which can be represented by its transverse line. In this case the problem is attributed to the contact between surface and curve.

scholarly journals Rapid Prototyping of Inertial MEMS Devices through Structural Optimization

Sensors ◽  
2021 ◽  
Vol 21 (15) ◽  
pp. 5064
Author(s):  
Daniele Giannini ◽  
Giacomo Bonaccorsi ◽  
Francesco Braghin
Keyword(s):  
Rigid Bodies ◽  
Design Tool ◽  
Mems Devices ◽  
Efficient Design ◽  
Computationally Efficient ◽  
Design Environment ◽  
Parametric Generation ◽  
Design And Optimization ◽  
Gradient Based ◽  
Complex Models

In this paper, we propose a novel design and optimization environment for inertial MEMS devices based on a computationally efficient schematization of the structure at the a device level. This allows us to obtain a flexible and efficient design optimization tool, particularly useful for rapid device prototyping. The presented design environment—feMEMSlite—handles the parametric generation of the structure geometry, the simulation of its dynamic behavior, and a gradient-based layout optimization. The methodology addresses the design of general inertial MEMS devices employing suspended proof masses, in which the focus is typically on the dynamics associated with the first vibration modes. In particular, the proposed design tool is tested on a triaxial beating-heart MEMS gyroscope, an industrially relevant and adequately complex example. The sensor layout is schematized by treating the proof masses as rigid bodies, discretizing flexural springs by Timoshenko beam finite elements, and accounting for electrostatic softening effects by additional negative spring constants. The MEMS device is then optimized according to two possible formulations of the optimization problem, including typical design requirements from the MEMS industry, with particular focus on the tuning of the structural eigenfrequencies and on the maximization of the response to external angular rates. The validity of the proposed approach is then assessed through a comparison with full FEM schematizations: rapidly prototyped layouts at the device level show a good performance when simulated with more complex models and therefore require only minor adjustments to accomplish the subsequent physical-level design.

Attractivity for differential equations of fractional order and ψ-Hilfer type

2020 ◽  
Vol 23 (4) ◽  
pp. 1188-1207
Author(s):  
J. Vanterler da C. Sousa ◽  
Mouffak Benchohra ◽  
Gaston M. N’Guérékata
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Fractional Derivatives ◽  
Broad Class ◽  
Hilfer Fractional Derivative ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Fractional Differential

AbstractThis paper investigates the overall solution attractivity of the fractional differential equation involving the ψ-Hilfer fractional derivative and using the Krasnoselskii’s fixed point theorem. We highlight some particular cases of the results presented here, especially involving the Riemann-Liouville, thus illustrating the broad class of fractional derivatives to which these results can be applied.

scholarly journals On Differential Properties Rν-Generalized Solution of the Dirichlet Problem with Coordinated Degeneration of the Input Data

2011 ◽  
Vol 2011 ◽  
pp. 1-18 ◽  
Author(s):  
Victor Anatolievich Rukavishnikov
Keyword(s):  
Differential Equation ◽  
Dirichlet Problem ◽  
Boundary Value Problem ◽  
Input Data ◽  
Order Differential Equation ◽  
Boundary Value ◽  
Generalized Solution ◽  
Strong Singularity ◽  
Second Order Differential Equation ◽  
Differential Properties

We consider the first boundary value problem for second-order differential equation with strong singularity caused by coordinated degeneration of the input data. For this problem, we study the differential properties of the Rν-generalized solution, that is, the fact that it belongs to the space H2,ν+β/2k+2(Ω).

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