233 research outputs found
Recommended from our members
Construction of periodic adapted orthonormal frames on closed space curves
The construction of continuous adapted orthonormal frames along C1 closedโloop spatial curves is addressed. Such frames are important in the design of periodic spatial rigidโbody motions along smooth closed paths. The construction is illustrated through the simplest nonโtrivial context โ namely, C1 closed loops defined by a single Pythagoreanโhodograph (PH) quintic space curve of a prescribed total arc length. It is shown that such curves comprise a twoโparameter family, dependent on two angular variables, and they degenerate to planar curves when these parameters differ by an integer multiple of ฯ. The desired frame is constructed through a rotation applied to the normalโplane vectors of the EulerโRodrigues frame, so as to interpolate a given initial/final frame orientation. A general solution for periodic adapted frames of minimal twist on C1 closedโloop PH curves is possible, although this incurs transcendental terms. However, the C1 closedโloop PH quintics admit particularly simple rational periodic adapted frames
Characterization of Spherical and Plane Curves Using Rotation Minimizing Frames
In this work, we study plane and spherical curves in Euclidean and
Lorentz-Minkowski 3-spaces by employing rotation minimizing (RM) frames. By
conveniently writing the curvature and torsion for a curve on a sphere, we show
how to find the angle between the principal normal and an RM vector field for
spherical curves. Later, we characterize plane and spherical curves as curves
whose position vector lies, up to a translation, on a moving plane spanned by
their unit tangent and an RM vector field. Finally, as an application, we
characterize Bertrand curves as curves whose so-called natural mates are
spherical.Comment: 8 pages. This version is an improvement of the previous one. In
addition to a study of some properties of plane and spherical curves, it
contains a characterization of Bertrand curves in terms of the so-called
natural mate
Recommended from our members
Mapping rational rotation-minimizing frames from polynomial curves on to rational curves
Given a polynomial space curve r(ฮพ) that has a rational rotationโminimizing frame (an RRMF curve), a methodology is developed to construct families of rational space curves rห(ฮพ) with the same rotationโminimizing frame as r(ฮพ) at corresponding points. The construction employs the dual form of a rational space curve, interpreted as the edge of regression of the envelope of a family of osculating planes, having normals in the direction u(ฮพ)=rโฒ(ฮพ)รrโณ(ฮพ) and distances from the origin specified in terms of a rational function f(ฮพ) as f(ฮพ)/โu(ฮพ)โ. An explicit characterization of the rational curves rห(ฮพ) generated by a given RRMF curve r(ฮพ) in this manner is developed, and the problem of matching initial and final points and frames is shown to impose only linear conditions on the coefficients of f(ฮพ), obviating the nonโlinear equations (and existence questions) that arise in addressing this problem with the RRMF curve r(ฮพ). Criteria for identifying lowโdegree instances of the curves rห(ฮพ) are identified, by a cancellation of factors common to their numerators and denominators, and the methodology is illustrated by a number of computed examples
๊ธฐํํ์ ์ผ๋ก ์ ๋ฐํ ๋น์ ํ ๊ตฌ์กฐ๋ฌผ์ ์์ด์-์ง์ค๋ฉํธ๋ฆญ ํ์ ์ค๊ณ ๋ฏผ๊ฐ๋ ํด์
ํ์๋
ผ๋ฌธ (๋ฐ์ฌ)-- ์์ธ๋ํ๊ต ๋ํ์ : ๊ณต๊ณผ๋ํ ์กฐ์ ํด์๊ณตํ๊ณผ, 2019. 2. ์กฐ์ ํธ.In this thesis, a continuum-based analytical adjoint configuration design sensitivity analysis (DSA) method is developed for gradient-based optimal design of curved built-up structures undergoing finite deformations. First, we investigate basic invariance property of linearized strain measures of a planar Timoshenko beam model which is combined with the selective reduced integration and B-bar projection method to alleviate shear and membrane locking. For a nonlinear structural analysis, geometrically exact beam and shell structural models are basically employed. A planar Kirchhoff beam problem is solved using the rotation-free discretization capability of isogeometric analysis (IGA) due to higher order continuity of NURBS basis function whose superior per-DOF(degree-of-freedom) accuracy over the conventional finite element analysis using Hermite basis function is verified. Various inter-patch continuity conditions including rotation continuity are enforced using Lagrage multiplier and penalty methods. This formulation is combined with a phenomenological constitutive model of shape memory polymer (SMP), and shape programming and recovery processes of SMP structures are simulated. Furthermore, for shear-deformable structures, a multiplicative update of finite rotations by an exponential map of a skew-symmetric matrix is employed. A procedure of explicit parameterization of local orthonormal frames in a spatial curve is presented using the smallest rotation method within the IGA framework. In the configuration DSA, the material derivative is applied to a variational equation, and an orientation design variation of curved structure is identified as a change of embedded local orthonormal frames. In a shell model, we use a regularized variational equation with a drilling rotational DOF. The material derivative of the orthogonal transformation matrix can be evaluated at final equilibrium configuration, which enables to compute design sensitivity using the tangent stiffness at the equilibrium without further iterations. A design optimization method for a constrained structure in a curved domain is also developed, which focuses on a lattice structure design on a specified surface. We define a lattice structure and its design variables on a rectangular plane, and utilize a concept of free-form deformation and a global curve interpolation to obtain an analytical expression for the control net of the structure on curved surface. The material derivative of the analytical expression eventually leads to precise design velocity field. Using this method, the number of design variables is reduced and design parameterization becomes more straightforward. In demonstrative examples, we verify the developed analytical adjoint DSA method in beam and shell structural problems undergoing finite deformations with various kinematic and force boundary conditions. The method is also applied to practical optimal design problems of curved built-up structures. For example, we extremize auxeticity of lattice structures, and experimentally verify nearly constant negative Poisson's ratio during large tensile and compressive deformations by using the 3-D printing and optical deformation measurement technologies. Also, we architect phononic band gap structures having significantly large band gap for mitigating noise in low audible frequency ranges.๋ณธ ์ฐ๊ตฌ์์๋ ๋๋ณํ์ ๊ณ ๋ คํ ํ์ด์ง ์กฐ๋ฆฝ ๊ตฌ์กฐ๋ฌผ์ ์ฐ์์ฒด ๊ธฐ๋ฐ ํด์์ ์ ์กฐ์ธ ํ์ ์ค๊ณ ๋ฏผ๊ฐ๋ ํด์ ๊ธฐ๋ฒ์ ๊ฐ๋ฐํ์๋ค. ํ๋ฉด Timoshenko ๋น์ ์ ํํ๋ ๋ณํ๋ฅ ์ invariance ํน์ฑ์ ๊ณ ์ฐฐํ์๊ณ invariant ์ ์ํ๋ฅผ ์ ํ์ ์ถ์์ ๋ถ(selective reduced integration) ๊ธฐ๋ฒ ๋ฐ B-bar projection ๊ธฐ๋ฒ๊ณผ ๊ฒฐํฉํ์ฌ shear ๋ฐ membrane ์ ๊น ํ์์ ํด์ํ์๋ค. ๋น์ ํ ๊ตฌ์กฐ ๋ชจ๋ธ๋ก์ ๊ธฐํํ์ ์ผ๋ก ์ ๋ฐํ ๋น ๋ฐ ์ ๋ชจ๋ธ์ ํ์ฉํ์๋ค. ํ๋ฉด Kirchhoff ๋น ๋ชจ๋ธ์ NURBS ๊ธฐ์ ํจ์์ ๊ณ ์ฐจ ์ฐ์์ฑ์ ๋ฐ๋ฅธ ์์ด์-์ง์ค๋ฉํธ๋ฆญ ํด์ ๊ธฐ๋ฐ rotation-free ์ด์ฐํ๋ฅผ ํ์ฉํ์ฌ ๋ค๋ฃจ์์ผ๋ฉฐ, ๊ธฐ์กด์ Hermite ๊ธฐ์ ํจ์ ๊ธฐ๋ฐ์ ์ ํ์์๋ฒ์ ๋นํด ์์ ๋๋น ํด์ ์ ํ๋๊ฐ ๋์์ ๊ฒ์ฆํ์๋ค. ๋ผ๊ทธ๋์ง ์น์๋ฒ ๋ฐ ๋ฒ์น ๊ธฐ๋ฒ์ ๋์
ํ์ฌ ํ์ ์ ์ฐ์์ฑ์ ํฌํจํ ๋ค์ํ ๋ค์คํจ์น๊ฐ ์ฐ์ ์กฐ๊ฑด์ ๊ณ ๋ คํ์๋ค. ์ด๋ฌํ ๊ธฐ๋ฒ์ ํ์ํ์ (phenomenological) ํ์๊ธฐ์ตํด๋ฆฌ๋จธ (SMP) ์ฌ๋ฃ ๊ตฌ์ฑ๋ฐฉ์ ์๊ณผ ๊ฒฐํฉํ์ฌ ํ์์ ํ๋ก๊ทธ๋๋ฐ๊ณผ ํ๋ณต ๊ณผ์ ์ ์๋ฎฌ๋ ์ด์
ํ์๋ค. ์ ๋จ๋ณํ์ ๊ฒช๋ (shear-deformable) ๊ตฌ์กฐ ๋ชจ๋ธ์ ๋ํ์ฌ ๋ํ์ ์ ๊ฐฑ์ ์ ๊ต๋ ํ๋ ฌ์ exponential map์ ์ํ ๊ณฑ์ ํํ๋ก ์ํํ์๋ค. ๊ณต๊ฐ์์ ๊ณก์ ๋ชจ๋ธ์์ ์ต์ํ์ (smallest rotation) ๊ธฐ๋ฒ์ ํตํด ๊ตญ์ ์ ๊ท์ง๊ต์ขํ๊ณ์ ๋ช
์์ ๋งค๊ฐํ๋ฅผ ์ํํ์๋ค. ํ์ ์ค๊ณ ๋ฏผ๊ฐ๋ ํด์์ ์ํ์ฌ ์ ๋ฏธ๋ถ์ ๋ณ๋ถ ๋ฐฉ์ ์์ ์ ์ฉํ์์ผ๋ฉฐ ํ์ด์ง ๊ตฌ์กฐ๋ฌผ์ ๋ฐฐํฅ ์ค๊ณ ๋ณํ๋ ๊ตญ์ ์ ๊ท์ง๊ต์ขํ๊ณ์ ํ์ ์ ์ํ์ฌ ๊ธฐ์ ๋๋ค. ์ต์ข
๋ณํ ํ์์์ ์ง๊ต ๋ณํ ํ๋ ฌ์ ์ ๋ฏธ๋ถ์ ๊ณ์ฐํจ์ผ๋ก์จ ๋ํ์ ๋ฌธ์ ์์ ์ถ๊ฐ์ ์ธ ๋ฐ๋ณต ๊ณ์ฐ์์ด ๋ณํ ํด์์์์ ์ ์ ๊ฐ์ฑํ๋ ฌ์ ์ํด ํด์์ ์ค๊ณ ๋ฏผ๊ฐ๋๋ฅผ ๊ณ์ฐํ ์ ์๋ค. ์ ๊ตฌ์กฐ๋ฌผ์ ๊ฒฝ์ฐ ๋ฉด๋ด ํ์ ์์ ๋ ๋ฐ ์์ ํ๋ ๋ณ๋ถ ๋ฐฉ์ ์์ ํ์ฉํ์ฌ ๋ณด๊ฐ์ฌ(stiffener)์ ๋ชจ๋ธ๋ง์ ์ฉ์ดํ๊ฒ ํ์๋ค. ๋ํ ๋ณธ ์ฐ๊ตฌ์์๋ ํ์ด์ง ์์ญ์ ๊ตฌ์๋์ด์๋ ๊ตฌ์กฐ๋ฌผ์ ๋ํ ์ค๊ณ ์๋์ฅ ๊ณ์ฐ ๋ฐ ์ต์ ์ค๊ณ๊ธฐ๋ฒ์ ์ ์ํ๋ฉฐ ํนํ ๊ณก๋ฉด์ ๊ตฌ์๋ ๋น ๊ตฌ์กฐ๋ฌผ์ ์ค๊ณ๋ฅผ ์ง์ค์ ์ผ๋ก ๋ค๋ฃฌ๋ค. ์์ ํ์๋ณํ(Free-form deformation)๊ธฐ๋ฒ๊ณผ ์ ์ญ ๊ณก์ ๋ณด๊ฐ๊ธฐ๋ฒ์ ํ์ฉํ์ฌ ์ง์ฌ๊ฐ ํ๋ฉด์์ ํ์ ๋ฐ ์ค๊ณ ๋ณ์๋ฅผ ์ ์ํ๊ณ ๊ณก๋ฉด์์ ๊ณก์ ํ์์ ๋ํ๋ด๋ ์กฐ์ ์ ์์น๋ฅผ ํด์์ ์ผ๋ก ํํํ ์ ์์ผ๋ฉฐ ์ด์ ์ ๋ฏธ๋ถ์ ํตํด ์ ํํ ์ค๊ณ์๋์ฅ์ ๊ณ์ฐํ๋ค. ์ด๋ฅผ ํตํด ์ค๊ณ ๋ณ์์ ๊ฐ์๋ฅผ ์ค์ผ ์ ์๊ณ ์ค๊ณ์ ๋งค๊ฐํ๊ฐ ๊ฐํธํด์ง๋ค. ๊ฐ๋ฐ๋ ๋ฐฉ๋ฒ๋ก ์ ๋ค์ํ ํ์ค ๋ฐ ์ด๋ํ์ ๊ฒฝ๊ณ์กฐ๊ฑด์ ๊ฐ๋ ๋น๊ณผ ์์ ๋๋ณํ ๋ฌธ์ ๋ฅผ ํตํด ๊ฒ์ฆ๋๋ฉฐ ์ฌ๋ฌ๊ฐ์ง ํ์ด์ง ์กฐ๋ฆฝ ๊ตฌ์กฐ๋ฌผ์ ์ต์ ์ค๊ณ์ ์ ์ฉ๋๋ค. ๋ํ์ ์ผ๋ก, ์ ๋จ ๊ฐ์ฑ ๋ฐ ์ถฉ๊ฒฉ ํก์ ํน์ฑ๊ณผ ๊ฐ์ ๊ธฐ๊ณ์ ๋ฌผ์ฑ์น์ ๊ฐ์ ์ ์ํด ํ์ฉ๋๋ ์ค๊ทธ์ ํฑ (auxetic) ํน์ฑ์ด ๊ทน๋ํ๋ ๊ฒฉ์ ๊ตฌ์กฐ๋ฅผ ์ค๊ณํ๋ฉฐ ์ธ์ฅ ๋ฐ ์์ถ ๋๋ณํ ๋ชจ๋์์ ์ผ์ ํ ์์ ํฌ์์ก๋น๋ฅผ ๋ํ๋์ 3์ฐจ์ ํ๋ฆฐํ
๊ณผ ๊ดํ์ ๋ณํ ์ธก์ ๊ธฐ์ ์ ์ด์ฉํ์ฌ ์คํ์ ์ผ๋ก ๊ฒ์ฆํ๋ค. ๋ํ ์ฐ๋ฆฌ๋ ์์์ ์ ๊ฐ์ ์ํด ํ์ฉ๋๋ ๊ฐ์ฒญ ์ ์ฃผํ์ ์์ญ๋์์์ ๋ฐด๋๊ฐญ์ด ๊ทน๋ํ๋ ๊ฒฉ์ ๊ตฌ์กฐ๋ฅผ ์ ์ํ๋ค.Abstract
1. Introduction
2. Isogeometric analysis of geometrically exact nonlinear structures
3. Isogeometric confinguration DSA of geometrically exact nonlinear structures
4. Numerical examples
5. Conclusions and future works
A. Supplements to the geometrically exact Kirchhoff beam model
B. Supplements to the geometrically exact shear-deformable beam model
C. Supplements to the geometrically exact shear-deformable shell model
D. Supplements to the invariant formulations
E. Supplements to the geometric constraints in design optimization
F. Supplements to the design of auxetic structures
์ด๋กDocto
Patient-specific technology for in vivo assessment of 3-D spinal motion
One of the most common musculoskeletal problems affecting people is neck and low back pain. Traditional clinical diagnostic techniques such as fluoroscopic imaging or CT scans are limited due to their static and/or planar measurements which may not be able to capture all neurological pathologies. More advanced diagnostics have proven successful in assessing 3-D patient-specific spinal kinematics by combining a patient-specific 3-D spine model (CT or MRI) with bi-planar fluoroscopic imaging; however, custom, not clinically available advanced imaging equipment as well as an increase in radiation exposure is required to acquire a complete patient-specific spinal kinematic description. Hence, the purpose of this research was to develop a clinically viable bi-planar fluoroscopic imaging technique which acquires a complete patient-specific kinematic description of the spine with reduced radiation exposure.
Development of the proposed technique required evaluating the accuracy of 3-D kinematic interpolation techniques in reconstructing spinal kinematic data in order to reduce radiation exposure from bi-planar fluoroscopic diagnostic techniques. Several interpolation and sampling algorithms were evaluated in reconstructing cadaveric lumbar (L2-S1) flexion-extension motion data; ultimately, a new interpolation algorithm was proposed. Similarly, the success of the interpolation algorithm was evaluated in reconstructing spine-specific kinematic parameters.
Next, the interpolation algorithm was combined with a CT-based bi-planar fluoroscopic method. Accuracy of the proposed diagnostic technique was evaluated against previously validated work on an ex vivo optoelectronic 3-D kinematic assessment technique. Bi-planar fluoroscopic images were acquired during both flexion-extension and lateral bending motions of cadaveric cervical (C4-T1) and lumbar (L2-S1) spine. Registration of the bi-planar fluoroscopic images to the CT-based 3-D model was optimized using a gradient derived similarity function. Additionally, a stochastic approach, covariance matrix adaptive evolution strategy, was used as the optimizing function. The newly developed interpolation algorithm was used to reduce the sample size of the bi-planar fluoroscopic images which reduces radiation exposure. Experimental results illustrate the potential success of the technique, but ultimately improvements in registration and validation methods are needed before becoming clinically viable
Numerical Methods in Shape Spaces and Optimal Branching Patterns
The contribution of this thesis is twofold. The main part deals with numerical methods in the context of shape space analysis, where the shape space at hand is considered as a Riemannian manifold. In detail, we apply and extend the time-discrete geodesic calculus (established by Rumpf and Wirth [WBRS11, RW15]) to the space of discrete shells, i.e. triangular meshes with fixed connectivity. The essential building block is a variational time-discretization of geodesic curves, which is based on a local approximation of the squared Riemannian distance on the manifold. On physical shape spaces this approximation can be derived e.g. from a dissimilarity measure. The dissimilarity measure between two shell surfaces can naturally be defined as an elastic deformation energy capturing both membrane and bending distortions. Combined with a non-conforming discretization of a physically sound thin shell model the time-discrete geodesic calculus applied to the space of discrete shells is shown to be suitable to solve important problems in computer graphics and animation. To extend the existing calculus, we introduce a generalized spline functional based on the covariant derivative along a curve in shape space whose minimizers can be considered as Riemannian splines. We establish a corresponding time-discrete functional that fits perfectly into the framework of Rumpf and Wirth, and prove this discretization to be consistent. Several numerical simulations reveal that the optimization of the spline functionalโsubject to appropriate constraintsโcan be used to solve the multiple interpolation problem in shape space, e.g. to realize keyframe animation. Based on the spline functional, we further develop a simple regression model which generalizes linear regression to nonlinear shape spaces. Numerical examples based on real data from anatomy and botany show the capability of the model. Finally, we apply the statistical analysis of elastic shape spaces presented by Rumpf and Wirth [RW09, RW11] to the space of discrete shells. To this end, we compute a Frรฉchet mean within a class of shapes bearing highly nonlinear variations and perform a principal component analysis with respect to the metric induced by the Hessian of an elastic shell energy. The last part of this thesis deals with the optimization of microstructures arising e.g. at austenite-martensite interfaces in shape memory alloys. For a corresponding scalar problem, Kohn and Mรผller [KM92, KM94] proved existence of a minimizer and provided a lower and an upper bound for the optimal energy. To establish the upper bound, they studied a particular branching pattern generated by mixing two different martensite phases. We perform a finite element simulation based on subdivision surfaces that suggests a topologically different class of branching patterns to represent an optimal microstructure. Based on these observations we derive a novel, low dimensional family of patterns and showโnumerically and analyticallyโthat our new branching pattern results in a significantly better upper energy bound
- โฆ