Advanced Mathematical Methods for Scientists and Engineers

Mathematica--A System for Doing Mathematics by Computer.

In this paper we consider (hierarchical, La-grange)reduced basis approximation anda posteriori error estimation for linear functional outputs of affinely parametrized elliptic coercive partial differential equa-tions. The essential ingredients are (primal-dual)Galer-kin projection onto a low-dimensional space associated with a smooth “parametric manifold” - dimension re-duction; efficient and effective greedy sampling meth-ods for identification of optimal and numerically stable approximations - rapid convergence;a posteriori er-ror estimation procedures - rigorous and sharp bounds for the linear-functional outputs of interest; and Offine-Online computational decomposition strategies - min-imummarginal cost for high performance in the real-time/embedded (e.g., parameter-estimation, control)and many-query (e.g., design optimization, multi-model/ scale)contexts. We present illustrative results for heat conduction and convection-diffusion,inviscid flow, and linear elasticity; outputs include transport rates, added mass,and stress intensity factors.

/pdf/reduced-basis-approximation-and-a-posteriori-error-1csv7ta4m6.pdf

Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations

Introduction. 1. Nonlinear theories of elasticity of plates and shells 2. Nonlinear theories of doubly curved shells for conventional and advanced materials 3. Introduction to nonlinear dynamics 4. Vibrations of rectangular plates 5. Vibrations of empty and fluid-filled circular cylindrical 6. Reduced order models: proper orthogonal decomposition and nonlinear normal modes 7. Comparison of different shell theories for nonlinear vibrations and stability of circular cylindrical shells 8. Effect of boundary conditions on a large-amplitude vibrations of circular cylindrical shells 9. Vibrations of circular cylindrical panels with different boundary conditions 10. Nonlinear vibrations and stability of doubly-curved shallow-shells: isotropic and laminated materials 11. Meshless discretization of plates and shells of complex shapes by using the R-functions 12. Vibrations of circular plates and rotating disks 13. Nonlinear stability of circular cylindrical shells under static and dynamic axial loads 14. Nonlinear stability and vibrations of circular shells conveying flow 15. Nonlinear supersonic flutter of circular cylindrical shells with imperfections.

/pdf/nonlinear-vibrations-and-stability-of-shells-and-plates-4zhg0tiabh.pdf

Nonlinear Vibrations and Stability of Shells and Plates

Introduction to Ordinary Differential Equations * Linear Systems and Stability of Nonlinear Systems * Applications * Hyperbolic Theory * Continuation of Periodic Solutions * Homoclinic Orbits, Melnikov?s Method, and Chaos * Averaging * Local Bifurcation * References * Index.

http://www.math.ecnu.edu.cn/~zmwang/teaching/Advanced_ODE-2014-2015/1-Textbook_and_Reference_book/2-Ordinary%20Differential%20Equations%20with%20Applications.pdf

Ordinary Differential Equations with Applications

Simple mixed models are developed for the geometrically nonlinear analysis of shells. A total Lagrangian description of the shell deformation is used, and the analytical formulation is based on a form of the nonlinear shallow shell theory with the effects of transverse shear deformation and bending-extensional coupling included. The fundamental unknowns consist of eight stress resultants and five generalized displacements of the shell, and the element characteristic arrays are obtained by using the Hellinger-Reissner mixed variational principle. The polynomial interpolation (or shape) functions used in approximating the stress resultants are, in general, of different degree than those used for approximating the generalized displacements. The stress resultants are discontinuous at the element boundaries and are eliminated on the element level.

The equivalence and ‘near-equivalence’ between the mixed models developed herein and displacement models based on reduced/selective integration of both transverse shear and extensional energy terms is discussed. The use of reduction methods in conjunction with the mixed models is outlined and the advantages of mixed models over displacement models are delineated. Analytic expressions are derived for the rigid-body and spurious (or zero energy) models for the various mixed models and their equivalent displacement models. Also, the advantages of mixed models over equivalent displacement models are outlined. Numerical results are presented to demonstrate the high accuracy and effectiveness of the mixed models developed, and to compare their performance with other mixed models reported in the literature.

Mixed models and reduced/selective integration displacement models for nonlinear shell analysis

Maintenance of indeterminacy is fundamental to the generation of plant architecture and a central component of the plant life strategy. Indeterminacy in plants is a characteristic of shoot and root meristems, which must balance maintenance of indeterminacy with organogenesis. The Petunia hybrida HAIRY MERISTEM (HAM) gene, a member of the GRAS family of transcriptional regulators, promotes shoot indeterminacy by an undefined non-cell-autonomous signaling mechanism(s). Here, we report that Arabidopsis (Arabidopsis thaliana) mutants triply homozygous for knockout alleles in three Arabidopsis HAM orthologs (Atham1,2,3 mutants) exhibit loss of indeterminacy in both the shoot and root. In the shoot, the degree of penetrance of the loss-of-indeterminacy phenotype of Atham1,2,3 mutants varies among shoot systems, with arrest of the primary vegetative shoot meristem occurring rarely or never, secondary shoot meristems typically arresting prior to initiating organogenesis, and inflorescence and flower meristems exhibiting a phenotypic range extending from wild type (flowers) to meristem arrest preempting organogenesis (flowers and inflorescence). Atham1,2,3 mutants also exhibit aberrant shoot phyllotaxis, lateral organ abnormalities, and altered meristem morphology in functioning meristems of both rosette and inflorescence. Root meristems of Atham1,2,3 mutants are significantly smaller than in the wild type in both longitudinal and radial axes, a consequence of reduced rates of meristem cell division that culminate in root meristem arrest. Atham1,2,3 phenotypes are unlikely to reflect complete loss of HAM function, as a fourth, more distantly related Arabidopsis HAM homolog, AtHAM4, exhibits overlapping function with AtHAM1 and AtHAM2 in promoting shoot indeterminacy.

/pdf/arabidopsis-homologs-of-the-petunia-hairy-meristem-gene-are-6mc5flshww.pdf

Arabidopsis Homologs of the Petunia HAIRY MERISTEM Gene Are Required for Maintenance of Shoot and Root Indeterminacy

An equation reported by van der Pol (1926) in connection with relaxation-oscillations studies is considered. The equation contains the factor epsilon which can assume values in the range from zero to infinity. The period T(epsilon), or equivalently the frequency nu(epsilon) of the limit cycle has been studied. However, to date there has been little success in discovering the analytical structure of T(epsilon) as a function of epsilon. The present investigation has the objectives to present the Taylor series expansion of nu(epsilon), to locate the singularities which determine the radius of convergence of that expansion, to introduce a new damping variable in terms of which the expansion converges for all epsilon, to form a new expansion for the period T(epsilon) which improves the rate of convergence of the series, to attempt to 'complete' the series, and to compare the obtained results with the numerically determined values of T(epsilon) and with the asymptotic approximation valid for large epsilon.

Power Series Expansions for the Frequency and Period of the Limit Cycle of the Van Der Pol Equation

Computerized symbolic manipulation in structural mechanics—Progress and potential

A power series expansion in the damping parameter $\varepsilon $ of the limit cycle $U(t;\varepsilon )$ of the free van der Pol equation $\ddot U + \varepsilon (U^2 - 1)\dot U + U = 0$ is constructed and analyzed. Coefficients in the expansion are computed up to $O(\varepsilon ^{24} )$ in exact rational arithmetic using the symbolic manipulation system MACSYMA and up to $O(\varepsilon ^{163} )$ using a FORTRAN program. The series is analyzed using Pade approximants. The convergence of the series for the maximum amplitude of the limit cycle is limited by two pairs of complex conjugate singularities in the complex $\varepsilon $-plane. These singularities are the same as those which limit the convergence of the series expansion of the frequency of the limit cycle. A new expansion parameter is introduced which maps these singularities to infinity and leads to a new expansion for the amplitude which converges for all real values of $\varepsilon $. Amplitudes computed from this transformed series agree very we...

/pdf/perturbation-analysis-of-the-limit-cycle-of-the-free-van-der-2nzwkwcfjb.pdf

C. M. Andersen

Papers

Mixed models and reduced/selective integration displacement models for nonlinear shell analysis

Arabidopsis Homologs of the Petunia HAIRY MERISTEM Gene Are Required for Maintenance of Shoot and Root Indeterminacy

Power Series Expansions for the Frequency and Period of the Limit Cycle of the Van Der Pol Equation

Computerized symbolic manipulation in structural mechanics—Progress and potential

Perturbation analysis of the limit cycle of the free van der Pol equation