This document summarizes a numerical analysis of a mixed finite element method for Reissner-Mindlin plates. It presents the continuous and discrete formulations, including the model problem, variational formulation, finite element spaces, and properties like ellipticity and inf-sup conditions. Key results shown are the well-posedness of both the continuous and discrete problems, and stability estimates for the solutions that are independent of the mesh size and plate thickness.
[Vvedensky d.] group_theory,_problems_and_solution(book_fi.org)Dabe Milli
1) The document discusses problems related to group theory.
2) Problem 1 shows that the wave equation for light propagation is invariant under Lorentz transformations.
3) Problem 2 shows that the Schrodinger equation is invariant under a global phase change of the wavefunction, and uses Noether's theorem to show the conservation of probability.
Dual Gravitons in AdS4/CFT3 and the Holographic Cotton TensorSebastian De Haro
This document summarizes research on dual gravitons in AdS4/CFT3 and the holographic Cotton tensor. Key points:
- In AdS4/CFT3, both modes of the graviton are normalizable, allowing duality to interchange the boundary metric g(0) and stress tensor T.
- The Cotton tensor C, which maps a metric to its stress tensor, plays a special role as the "holographic Cotton tensor".
- There is a duality symmetry of the bulk equations of motion under which the linearized metric fluctuations ḣ and Cotton tensor of the dual metric C(ḣ) are interchanged.
- This relates the
This document summarizes Chris Swierczewski's general exam presentation on computational applications of Riemann surfaces and Abelian functions. The presentation covered the geometry and algebra of Riemann surfaces, including bases of cycles, holomorphic differentials, and period matrices. Applications discussed include using Riemann theta functions to find periodic solutions to integrable PDEs like the Kadomtsev–Petviashvili equation. The talk also discussed linear matrix representations of algebraic curves and the constructive Schottky problem of realizing a Riemann matrix as the period matrix of a curve.
This document contains solutions to exercises from a pre-calculus textbook on radical functions.
1) It provides tables, graphs and explanations for various radical functions such as √x, √x+3, and their relation to other functions.
2) Students are asked to sketch graphs of radical functions based on given quadratic, cubic or other functions, and identify domains and ranges.
3) Radical equations are solved by graphing related functions and finding the x-intercept(s).
Overall Viscoelastic Properties of Fiber-Reinforced Hierarchical CompositesOscar L. Cruz González
This document describes a three-scale asymptotic homogenization method for determining the overall viscoelastic properties of fiber-reinforced hierarchical composites.
The method involves separating the problem into three structural scales - the macroscale, mesoscale (ε1), and microscale (ε2). Local problems are solved at the mesoscale and microscale to determine the effective coefficients at each scale.
A three-dimensional computational approach is used at the mesoscale, where the microscale unit cell problem is simplified by assuming the material properties are piecewise constant over two phases. Solutions of the local problems provide the effective relaxation modulus at each scale, with the macroscale problem governing the overall viscoelastic response of the composite material.
This document provides definitions and formulas from theoretical computer science, including:
1. Big O, Omega, and Theta notation for analyzing algorithm complexity.
2. Common series like geometric and harmonic series.
3. Recurrence relations and methods for solving them like the master theorem.
4. Combinatorics topics like permutations, combinations, and binomial coefficients.
This document discusses using Gaussian process models for change point detection in atmospheric dispersion problems. It proposes using multiple kernels in a Gaussian process to model different regimes indicated by change points. A two-stage process is used to first estimate the change point (release time) and then estimate the source location. Simulation results show the approach outperforms existing techniques in estimating change points and source locations from concentration sensor measurements. The approach is applied to model real concentration data to estimate a CBRN release scenario.
Gauge Systems With Noncommutative Phase Spaceguest9fa195
This document describes gauge systems with noncommutative phase spaces. It introduces several models of gauge systems where the phase space has a noncanonical symplectic structure involving parameters that encode noncommutativity among coordinates and momenta.
As an example, it considers a noncommutative version of the usual SL(2,R) model where the symplectic structure is modified by a parameter θ that introduces noncommutativity between one set of coordinates. The constraints of the original model are also modified to maintain the same gauge algebra. The dynamics of this noncommutative SL(2,R) model involve additional terms depending on θ.
More generally, the paper shows it is possible to construct gauge systems where non
[Vvedensky d.] group_theory,_problems_and_solution(book_fi.org)Dabe Milli
1) The document discusses problems related to group theory.
2) Problem 1 shows that the wave equation for light propagation is invariant under Lorentz transformations.
3) Problem 2 shows that the Schrodinger equation is invariant under a global phase change of the wavefunction, and uses Noether's theorem to show the conservation of probability.
Dual Gravitons in AdS4/CFT3 and the Holographic Cotton TensorSebastian De Haro
This document summarizes research on dual gravitons in AdS4/CFT3 and the holographic Cotton tensor. Key points:
- In AdS4/CFT3, both modes of the graviton are normalizable, allowing duality to interchange the boundary metric g(0) and stress tensor T.
- The Cotton tensor C, which maps a metric to its stress tensor, plays a special role as the "holographic Cotton tensor".
- There is a duality symmetry of the bulk equations of motion under which the linearized metric fluctuations ḣ and Cotton tensor of the dual metric C(ḣ) are interchanged.
- This relates the
This document summarizes Chris Swierczewski's general exam presentation on computational applications of Riemann surfaces and Abelian functions. The presentation covered the geometry and algebra of Riemann surfaces, including bases of cycles, holomorphic differentials, and period matrices. Applications discussed include using Riemann theta functions to find periodic solutions to integrable PDEs like the Kadomtsev–Petviashvili equation. The talk also discussed linear matrix representations of algebraic curves and the constructive Schottky problem of realizing a Riemann matrix as the period matrix of a curve.
This document contains solutions to exercises from a pre-calculus textbook on radical functions.
1) It provides tables, graphs and explanations for various radical functions such as √x, √x+3, and their relation to other functions.
2) Students are asked to sketch graphs of radical functions based on given quadratic, cubic or other functions, and identify domains and ranges.
3) Radical equations are solved by graphing related functions and finding the x-intercept(s).
Overall Viscoelastic Properties of Fiber-Reinforced Hierarchical CompositesOscar L. Cruz González
This document describes a three-scale asymptotic homogenization method for determining the overall viscoelastic properties of fiber-reinforced hierarchical composites.
The method involves separating the problem into three structural scales - the macroscale, mesoscale (ε1), and microscale (ε2). Local problems are solved at the mesoscale and microscale to determine the effective coefficients at each scale.
A three-dimensional computational approach is used at the mesoscale, where the microscale unit cell problem is simplified by assuming the material properties are piecewise constant over two phases. Solutions of the local problems provide the effective relaxation modulus at each scale, with the macroscale problem governing the overall viscoelastic response of the composite material.
This document provides definitions and formulas from theoretical computer science, including:
1. Big O, Omega, and Theta notation for analyzing algorithm complexity.
2. Common series like geometric and harmonic series.
3. Recurrence relations and methods for solving them like the master theorem.
4. Combinatorics topics like permutations, combinations, and binomial coefficients.
This document discusses using Gaussian process models for change point detection in atmospheric dispersion problems. It proposes using multiple kernels in a Gaussian process to model different regimes indicated by change points. A two-stage process is used to first estimate the change point (release time) and then estimate the source location. Simulation results show the approach outperforms existing techniques in estimating change points and source locations from concentration sensor measurements. The approach is applied to model real concentration data to estimate a CBRN release scenario.
Gauge Systems With Noncommutative Phase Spaceguest9fa195
This document describes gauge systems with noncommutative phase spaces. It introduces several models of gauge systems where the phase space has a noncanonical symplectic structure involving parameters that encode noncommutativity among coordinates and momenta.
As an example, it considers a noncommutative version of the usual SL(2,R) model where the symplectic structure is modified by a parameter θ that introduces noncommutativity between one set of coordinates. The constraints of the original model are also modified to maintain the same gauge algebra. The dynamics of this noncommutative SL(2,R) model involve additional terms depending on θ.
More generally, the paper shows it is possible to construct gauge systems where non
Likelihood is sometimes difficult to compute because of the complexity of the model. Approximate Bayesian computation (ABC) makes it easy to sample parameters generating approximation of observed data.
The document discusses complex differentiability and analytic functions. It shows that for a function f(z) to be complex differentiable, the Cauchy-Riemann equations relating the partial derivatives of the real and imaginary parts must be satisfied. It also discusses representing functions as power series and their radii of convergence. Multivalued functions like logarithms and roots are discussed, noting the need for branch cuts to define single-valued branches.
1. The document provides definitions and identities relating to theoretical computer science topics like asymptotic analysis, series, recurrences, and discrete structures.
2. It includes definitions for big O, Omega, and Theta notation used to describe the asymptotic behavior of functions.
3. Recurrences, like the master method, are presented for analyzing the runtime of divide-and-conquer algorithms.
4. Discrete structures like combinations, Stirling numbers, and trees are also covered, along with their properties and relationships.
The document discusses equations of straight lines. It covers determining the gradient and equation of a straight line, as well as representing a line in various forms such as y=mx+c, Ax + By + C = 0, and identifying lines from two points. Key concepts covered include finding the gradient from two points, the relationship between perpendicular lines, and deriving the equation of a line given values for the gradient and a point.
Non-sampling functional approximation of linear and non-linear Bayesian UpdateAlexander Litvinenko
We offer a non-sampling functional approximation of non-linear surrogate to classical Bayesian Update formula. We start with prior Polynomial Chaos Expansion (PCE), express log-likelihood in a PCE basis and obtain a new posterior PCE.
Main IDEA is to update not probability density, but basis coefficients.
This document contains the work of a student on a calculus test. It includes:
1) Solving limits, finding derivatives, and applying L'Hopital's rule.
2) Using induction to prove an identity.
3) Providing epsilon-delta proofs of limits.
4) Finding where a tangent line is parallel to a secant line.
5) Proving statements about limits of functions.
The student provides detailed solutions showing their work for each problem on the test.
This document discusses trapped modes and scattering problems for an infinite nonhomogeneous Timoshenko beam. It formulates the problem using a Timoshenko system and introduces non-dimensional quantities. It derives an integral equation for the Green matrix and discusses obtaining a solution that decreases at infinity by requiring coefficients of oscillating terms equal zero. It finds a solution under the condition that the integral of the perturbation is positive and requiring the perturbation is even to obtain a single equation.
This document discusses solving quadratic equations. It begins by defining quadratic equations as equations of the second degree in the form ax2 + bx + c = 0, where a ≠ 0. It then discusses:
- The roots or solutions of a quadratic equation are the values that make the equation equal to 0 when substituted for the variable.
- Quadratic equations can be solved by factorization, splitting the middle term into two factors and setting each factor equal to 0 to find the roots.
- Examples are provided to demonstrate solving quadratic equations by factorization, finding the two roots.
- A more general quadratic equation with parameters a, b is also factorized to find its two equal roots.
1) Euler-Bernoulli bending theory and Timoshenko beam theory describe the stresses and deflections of beams under bending loads.
2) Euler-Bernoulli theory assumes a beam's cross-section remains plane and perpendicular to the neutral axis during bending. Timoshenko theory accounts for shear deformation.
3) Both theories relate the bending moment M and shear force V to the beam's deflection w and its derivatives, allowing calculation of stresses, forces, and deflections for given beam geometries and loads.
This document contains an unsolved mathematics paper from 2004 containing 37 multiple choice problems testing critical reasoning skills. Some example problems include finding the digit sum of an arithmetic expression, determining the angle of intersection of two curves, and finding the value of x that satisfies a complex logarithmic equation. The problems cover a wide range of mathematics topics including algebra, trigonometry, logarithms, and geometry.
This document summarizes key concepts in unconstrained optimization of functions with two variables, including:
1) Critical points are found by taking the partial derivatives and setting them equal to zero, generalizing the first derivative test for single-variable functions.
2) The Hessian matrix generalizes the second derivative, with its entries being the partial derivatives evaluated at a critical point.
3) The second derivative test classifies critical points as local maxima, minima or saddle points based on the signs of the Hessian matrix's eigenvalues.
4) Taylor polynomial approximations in two variables involve partial derivatives up to second order, analogous to single-variable Taylor series.
5) An example classifies the critical points
This document contains the solutions to Exercise 7.4 from the Textbook of Algebra and Trigonometry for Class XI. It includes solutions to 9 questions involving combinations and permutations. The questions cover topics such as calculating factorials, binomial coefficients, and determining the number of combinations and permutations based on given parameters. Detailed step-by-step workings are shown for each question.
Cosmin Crucean: Perturbative QED on de Sitter Universe.SEENET-MTP
The document summarizes key aspects of quantum field theory on de Sitter spacetime, including solutions to the Dirac, scalar, electromagnetic, and other field equations. It presents:
1) Fundamental solutions for the Dirac equation and orthonormalization relations for Dirac spinor modes.
2) Solutions to the Klein-Gordon equation for a scalar field and corresponding orthonormalization relations.
3) Quantization of electromagnetic vector potentials in the Coulomb gauge and orthonormalization relations for photon modes.
The document summarizes two models:
1. The Lo-Zivot Threshold Cointegration Model, which uses a threshold vector error correction model (TVECM) to analyze the dynamic adjustment of cointegrated time series variables to their long-run equilibrium. It allows for nonlinear and asymmetric adjustment speeds.
2. A bivariate vector error correction model (VECM) and band-threshold vector error correction model (BAND-TVECM) that extend the VECM to allow for nonlinear and discontinuous adjustments to long-run equilibrium across multiple regimes defined by thresholds on a variable. This captures asymmetric adjustment speeds and dynamic behavior.
The BAND-TVECM allows modeling of
Complex analysis and differential equationSpringer
This document introduces holomorphic functions and some of their key properties. It begins by defining limits, continuity, differentiability, and holomorphic functions. It then introduces the Cauchy-Riemann equations, which provide a necessary condition for differentiability involving the partial derivatives of the real and imaginary parts. Several examples are provided to illustrate these concepts. The document also discusses properties of derivatives of holomorphic functions and proves that differentiability implies continuity. It concludes by defining connected sets.
Quantum fields on the de sitter spacetime - Ion CotaescuSEENET-MTP
This document summarizes research on defining quantum fields on de Sitter spacetime. It discusses how external symmetries allow defining conserved observables like an energy operator, despite doubts in the literature. New quantum modes were obtained for scalar, Dirac, and vector fields on de Sitter spacetime using this energy operator. The paper reviews defining fields in local frames where spin is well-defined, and how isometries give rise to conserved quantities through external symmetry transformations that involve gauge transformations and diffeomorphisms. Generators of field representations are constructed from orbital and spin parts related to Killing vectors and structure functions.
This document discusses differential equations. It begins by explaining that differential equations are used to model many physical phenomena in areas like economics, engineering, and more. It then provides examples of ordinary and partial differential equations. The rest of the document defines key terms related to differential equations like order, degree, families of curves, and how to derive the differential equation of a family of curves by eliminating parameters. Several examples are provided to illustrate these concepts.
The document provides 4 examples of calculating triple integrals over different regions. Example 1 calculates a triple integral over a region bounded by a paraboloid and plane in rectangular coordinates, then evaluates it in polar coordinates. Example 2 calculates a triple integral over a hemisphere in spherical coordinates. Example 3 finds the volume under a paraboloid and above a rectangle using a double integral. Example 4 calculates a triple integral over a tetrahedron bounded by 4 planes.
This document provides definitions and properties for trigonometric functions including sine, cosine, and tangent. It defines the domains and ranges of sine, cosine, and tangent. Examples of trigonometric ratios are given for common angles like 30, 45, 60, and 90 degrees. Trigonometric identities are also listed, such as the sine and cosine of sums and differences of angles.
Likelihood is sometimes difficult to compute because of the complexity of the model. Approximate Bayesian computation (ABC) makes it easy to sample parameters generating approximation of observed data.
The document discusses complex differentiability and analytic functions. It shows that for a function f(z) to be complex differentiable, the Cauchy-Riemann equations relating the partial derivatives of the real and imaginary parts must be satisfied. It also discusses representing functions as power series and their radii of convergence. Multivalued functions like logarithms and roots are discussed, noting the need for branch cuts to define single-valued branches.
1. The document provides definitions and identities relating to theoretical computer science topics like asymptotic analysis, series, recurrences, and discrete structures.
2. It includes definitions for big O, Omega, and Theta notation used to describe the asymptotic behavior of functions.
3. Recurrences, like the master method, are presented for analyzing the runtime of divide-and-conquer algorithms.
4. Discrete structures like combinations, Stirling numbers, and trees are also covered, along with their properties and relationships.
The document discusses equations of straight lines. It covers determining the gradient and equation of a straight line, as well as representing a line in various forms such as y=mx+c, Ax + By + C = 0, and identifying lines from two points. Key concepts covered include finding the gradient from two points, the relationship between perpendicular lines, and deriving the equation of a line given values for the gradient and a point.
Non-sampling functional approximation of linear and non-linear Bayesian UpdateAlexander Litvinenko
We offer a non-sampling functional approximation of non-linear surrogate to classical Bayesian Update formula. We start with prior Polynomial Chaos Expansion (PCE), express log-likelihood in a PCE basis and obtain a new posterior PCE.
Main IDEA is to update not probability density, but basis coefficients.
This document contains the work of a student on a calculus test. It includes:
1) Solving limits, finding derivatives, and applying L'Hopital's rule.
2) Using induction to prove an identity.
3) Providing epsilon-delta proofs of limits.
4) Finding where a tangent line is parallel to a secant line.
5) Proving statements about limits of functions.
The student provides detailed solutions showing their work for each problem on the test.
This document discusses trapped modes and scattering problems for an infinite nonhomogeneous Timoshenko beam. It formulates the problem using a Timoshenko system and introduces non-dimensional quantities. It derives an integral equation for the Green matrix and discusses obtaining a solution that decreases at infinity by requiring coefficients of oscillating terms equal zero. It finds a solution under the condition that the integral of the perturbation is positive and requiring the perturbation is even to obtain a single equation.
This document discusses solving quadratic equations. It begins by defining quadratic equations as equations of the second degree in the form ax2 + bx + c = 0, where a ≠ 0. It then discusses:
- The roots or solutions of a quadratic equation are the values that make the equation equal to 0 when substituted for the variable.
- Quadratic equations can be solved by factorization, splitting the middle term into two factors and setting each factor equal to 0 to find the roots.
- Examples are provided to demonstrate solving quadratic equations by factorization, finding the two roots.
- A more general quadratic equation with parameters a, b is also factorized to find its two equal roots.
1) Euler-Bernoulli bending theory and Timoshenko beam theory describe the stresses and deflections of beams under bending loads.
2) Euler-Bernoulli theory assumes a beam's cross-section remains plane and perpendicular to the neutral axis during bending. Timoshenko theory accounts for shear deformation.
3) Both theories relate the bending moment M and shear force V to the beam's deflection w and its derivatives, allowing calculation of stresses, forces, and deflections for given beam geometries and loads.
This document contains an unsolved mathematics paper from 2004 containing 37 multiple choice problems testing critical reasoning skills. Some example problems include finding the digit sum of an arithmetic expression, determining the angle of intersection of two curves, and finding the value of x that satisfies a complex logarithmic equation. The problems cover a wide range of mathematics topics including algebra, trigonometry, logarithms, and geometry.
This document summarizes key concepts in unconstrained optimization of functions with two variables, including:
1) Critical points are found by taking the partial derivatives and setting them equal to zero, generalizing the first derivative test for single-variable functions.
2) The Hessian matrix generalizes the second derivative, with its entries being the partial derivatives evaluated at a critical point.
3) The second derivative test classifies critical points as local maxima, minima or saddle points based on the signs of the Hessian matrix's eigenvalues.
4) Taylor polynomial approximations in two variables involve partial derivatives up to second order, analogous to single-variable Taylor series.
5) An example classifies the critical points
This document contains the solutions to Exercise 7.4 from the Textbook of Algebra and Trigonometry for Class XI. It includes solutions to 9 questions involving combinations and permutations. The questions cover topics such as calculating factorials, binomial coefficients, and determining the number of combinations and permutations based on given parameters. Detailed step-by-step workings are shown for each question.
Cosmin Crucean: Perturbative QED on de Sitter Universe.SEENET-MTP
The document summarizes key aspects of quantum field theory on de Sitter spacetime, including solutions to the Dirac, scalar, electromagnetic, and other field equations. It presents:
1) Fundamental solutions for the Dirac equation and orthonormalization relations for Dirac spinor modes.
2) Solutions to the Klein-Gordon equation for a scalar field and corresponding orthonormalization relations.
3) Quantization of electromagnetic vector potentials in the Coulomb gauge and orthonormalization relations for photon modes.
The document summarizes two models:
1. The Lo-Zivot Threshold Cointegration Model, which uses a threshold vector error correction model (TVECM) to analyze the dynamic adjustment of cointegrated time series variables to their long-run equilibrium. It allows for nonlinear and asymmetric adjustment speeds.
2. A bivariate vector error correction model (VECM) and band-threshold vector error correction model (BAND-TVECM) that extend the VECM to allow for nonlinear and discontinuous adjustments to long-run equilibrium across multiple regimes defined by thresholds on a variable. This captures asymmetric adjustment speeds and dynamic behavior.
The BAND-TVECM allows modeling of
Complex analysis and differential equationSpringer
This document introduces holomorphic functions and some of their key properties. It begins by defining limits, continuity, differentiability, and holomorphic functions. It then introduces the Cauchy-Riemann equations, which provide a necessary condition for differentiability involving the partial derivatives of the real and imaginary parts. Several examples are provided to illustrate these concepts. The document also discusses properties of derivatives of holomorphic functions and proves that differentiability implies continuity. It concludes by defining connected sets.
Quantum fields on the de sitter spacetime - Ion CotaescuSEENET-MTP
This document summarizes research on defining quantum fields on de Sitter spacetime. It discusses how external symmetries allow defining conserved observables like an energy operator, despite doubts in the literature. New quantum modes were obtained for scalar, Dirac, and vector fields on de Sitter spacetime using this energy operator. The paper reviews defining fields in local frames where spin is well-defined, and how isometries give rise to conserved quantities through external symmetry transformations that involve gauge transformations and diffeomorphisms. Generators of field representations are constructed from orbital and spin parts related to Killing vectors and structure functions.
This document discusses differential equations. It begins by explaining that differential equations are used to model many physical phenomena in areas like economics, engineering, and more. It then provides examples of ordinary and partial differential equations. The rest of the document defines key terms related to differential equations like order, degree, families of curves, and how to derive the differential equation of a family of curves by eliminating parameters. Several examples are provided to illustrate these concepts.
The document provides 4 examples of calculating triple integrals over different regions. Example 1 calculates a triple integral over a region bounded by a paraboloid and plane in rectangular coordinates, then evaluates it in polar coordinates. Example 2 calculates a triple integral over a hemisphere in spherical coordinates. Example 3 finds the volume under a paraboloid and above a rectangle using a double integral. Example 4 calculates a triple integral over a tetrahedron bounded by 4 planes.
This document provides definitions and properties for trigonometric functions including sine, cosine, and tangent. It defines the domains and ranges of sine, cosine, and tangent. Examples of trigonometric ratios are given for common angles like 30, 45, 60, and 90 degrees. Trigonometric identities are also listed, such as the sine and cosine of sums and differences of angles.
A Szemeredi-type theorem for subsets of the unit cubeVjekoslavKovac1
This document summarizes a talk on gaps between arithmetic progressions in subsets of the unit cube. It presents three key propositions:
1) For subsets A of positive measure, structured progressions contribute a lower bound depending on the measure of A and the best known bounds for Szemerédi's theorem.
2) Estimating errors by pigeonholing scales, the difference between smooth and sharp progressions over various scales is bounded above by a sublinear function of scales.
3) For sufficiently nice subsets, the difference between measure and smoothed measure is arbitrarily small by choosing a small smoothing parameter.
Combining these propositions shows that for sufficiently nice subsets, gaps between progressions contain an interval
The document discusses difference systems of sets (DSS), which are sets with certain distance properties. It provides examples of perfect and regular DSS, and discusses approaches to constructing optimal DSS, including using cyclic difference packings, flats in projective geometry, and cyclotomic classes. Optimal constructions are given using finite fields.
The document discusses difference systems of sets (DSS), which are sets with certain distance properties. It provides examples of perfect and regular DSS, and discusses approaches to constructing optimal DSS, including using cyclic difference packings, flats in projective geometry, and cyclotomic classes. Optimal constructions are given using finite fields and cyclotomic cosets.
This document provides an introduction to fundamental mathematics concepts including:
1) Algebra topics such as quadratic equations, binomial theorem, arithmetic and geometric progressions, and common formulae.
2) Trigonometry including trigonometric ratios, values of ratios for standard angles, fundamental relations, and ratios of allied angles.
3) Examples are provided to demonstrate solving problems using these mathematical concepts and their application to physics principles. The document serves as an overview of basic but essential mathematics needed for understanding and applying physical principles.
The document discusses Legendre polynomials, which are special functions that arise in solutions to Laplace's equation in spherical coordinates. Some key points:
1) Legendre polynomials Pn(cosθ) are a set of orthogonal polynomials that satisfy Legendre's differential equation.
2) Pn(cosθ) can be defined using a generating function or by taking partial derivatives of 1/r.
3) Important properties of Legendre polynomials include P0(t)=1, Pn(1)=1, Pn(-1)=(-1)n, and a recurrence relation involving Pn+1, Pn, and their derivatives.
This document contains exam questions related to Engineering Mathematics and Microcontrollers.
Part A of Engineering Mathematics asks students to: 1) Find an approximate value of y at x=0.1 and 0.2 using Taylor's series, 2) Solve a differential equation using Euler's modified method and carry out three modifications, 3) Determine the value of y(1.4) using Adams-Bashforth method given values of y at other points.
Part B asks students to: 1) Fit a least squares line to given data, 2) Prove and explain a trigonometric identity, 3) Find the probability of solving a problem given individual student probabilities, 4) Define terms related to probability distributions,
This document contains instructions for 5 assignment questions involving numerical integration and solving differential equations. Question 1 involves using the quad function to evaluate several integrals. Question 2 involves using quad to evaluate Fresnel integrals and plot the results. Question 3 involves using Monte Carlo methods to estimate volumes and double integrals. Question 4 involves using Euler's method to solve an initial value problem and analyze errors. Question 5 involves using lsode to solve a system of differential equations modeling atmospheric circulation and experimenting with initial conditions.
The document summarizes research on magnetic monopoles in noncommutative spacetime. It begins by motivating noncommutative spacetime as a way to incorporate quantum gravitational effects. It then shows that attempting to quantize spacetime by imposing noncommutativity of coordinates leads to inconsistencies when trying to define a Wu-Yang magnetic monopole in this framework. Specifically, the potentials describing the monopole fail to simultaneously satisfy Maxwell's equations and transform correctly under gauge transformations when expanded to second order in the noncommutativity parameter. This suggests the Dirac quantization condition cannot be satisfied in noncommutative spacetime. Possible reasons for this failure and directions for future work are discussed.
Research Inventy : International Journal of Engineering and Scienceinventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
The document discusses efficient algorithms for performing approximate matching queries on strings that have been grammar-compressed. It introduces the concept of implicit unit-Monge matrices which can represent permutation matrices in a space-efficient way using a range tree data structure. This representation allows dominance counting queries, needed for string comparison, to be performed in O(log2 n) time after an O(n log n) preprocessing step. More advanced data structures can improve these asymptotic time and space bounds further.
The document discusses rational points on elliptic curves. It begins by introducing points on conic sections and how to parametrize them when a rational point exists. It then discusses elliptic curves, which have a group structure on their rational points. The Mordell-Weil theorem states the group of rational points is finitely generated. The document discusses counting points on elliptic curves over finite fields and relates this to L-functions via the Birch and Swinnerton-Dyer conjecture. It concludes by introducing Heegner points, which are used to study ranks of elliptic curves over number fields.
The document describes a damped mass-spring system and provides the equation of motion for analyzing the free vibration of the system. It then gives the general solution to the differential equation that describes the response x(t) in terms of the system's natural frequency, damping ratio, initial displacement, and initial velocity. The student is asked to:
1. Create a Matlab function to calculate the response x(t) for given parameter values.
2. Run sample code that plots the response for different damping ratios.
3. Calculate and submit the response at two specific cases.
This document provides an overview of nanomagnetism and the key developments in the field over time. It discusses early experiments and models from Einstein, de Haas, Heisenberg, and others. Key concepts covered include magnetic anisotropy, superparamagnetism, quantum tunneling of magnetization, and magnetic deflagration. Experimental work is highlighted from various researchers, including observations of quantum steps in microwave absorption and avalanches in Mn-12 acetate.
Change of variables in double integralsTarun Gehlot
1. The document discusses change of variables for double integrals, introducing the Jacobian determinant which relates the differentials of the original and transformed variables.
2. It provides an example of using a change of variables (u=x-y, v=x+y) to evaluate an integral over a parallelogram region.
3. Polar coordinates are also discussed as a common change of variables technique for double integrals, with an example evaluating an integral over a circular region in polar coordinates.
Change of variables in double integralsTarun Gehlot
1. The document discusses change of variables for double integrals, introducing the Jacobian determinant which relates the differentials of the original and transformed variables.
2. It provides an example of using a change of variables (u=x-y, v=x+y) to evaluate an integral over a parallelogram region.
3. Polar coordinates are also discussed as a common change of variables technique for double integrals, with an example evaluating an integral over a circular region in polar coordinates.
1. Numerical analysis of a locking-free mixed
fem for a bending moment formulation of
Reissner-Mindlin plates
¸ ˜
L OURENC O B EIR A O DA V EIGA1 , DAVID M ORA2 , R ODOLFO R ODR´GUEZ3
I
1
`
Dipartimento di Matematica, Universita degli Studi di Milano.
2
´
Departamento de Matematica, Universidad del B´o B´o.
ı ı
3
´ ´
Departamento de Ingenier´a Matematica, Universidad de Concepcion.
ı
´ ´ ´
Sesion Especial de Analisis Numerico
´
XX Congreso de Matematica Capricornio COMCA 2010
´
Universidad de Tarapaca, Arica, Chile
4–6 de Agosto de 2010.
2. Contents
• The model problem
• The continuous formulation
• Galerkin formulation
• Numerical tests
Mixed FEM for Reissner-Mindlin plates. –2– B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
˜ I
3. The model problem.
Given g ∈ L2 (Ω), find β , γ and w
− div(C(ε(β))) − γ = 0 in Ω,
−div γ = g in Ω,
κ
γ = 2 (∇w − β) in Ω,
t
w = 0, β = 0 on ∂Ω,
• w is the transverse displacement,
• β = (β1 , β2 ) are the rotations,
• γ is the shear stress,
• t is the thickness,
• κ := Ek/2(1 + ν) is the shear modulus (k is a correction factor),
E
• Cτ := 12(1−ν 2 ) ((1 − ν)τ + ν tr(τ )I) ,
• we restrict our analysis to convex plates.
Mixed FEM for Reissner-Mindlin plates. –3– B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
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4. The model problem. (cont.)
We introduce as a new unknown the bending moment σ = (σij )1≤i,j≤2 , defined by
σ := C(ε(β)).
12(1 − ν 2 ) 1 ν
• C −1 τ := τ− 2)
tr(τ )I .
E (1 − ν) (1 − ν
We rewrite the equation above as:
−1 1
C σ = ∇β − (∇β − (∇β)t ) = ∇β − rJ,
2
1
• r := − 2 rot β ,
0 1
• J := .
−1 0
Mixed FEM for Reissner-Mindlin plates. –4– B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
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5. The continuous formulation.
Continuous mixed problem
Find ((σ, γ), (β, r, w)) ∈ H × Q such that
t2
C −1 σ : τ + γ·ξ+ β · (div τ + ξ) + r(τ12 − τ21 ) + wdiv ξ = 0
Ω κ Ω Ω Ω Ω
η · (div σ + γ) + s(σ12 − σ21 ) + vdiv γ = − gv,
Ω Ω Ω Ω
for all ((τ , ξ), (η, s, v)) ∈ H × Q, where
H := H(div; Ω) × H(div ; Ω),
Q := [L2 (Ω)]2 × L2 (Ω) × L2 (Ω),
with
H(div; Ω) := {τ ∈ [L2 (Ω)]2×2 : div τ ∈ [L2 (Ω)]2 },
and
H(div ; Ω) := {ξ ∈ [L2 (Ω)]2 : div ξ ∈ L2 (Ω)}.
Mixed FEM for Reissner-Mindlin plates. –5– B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
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6. The continuous formulation. (cont.)
Continuous mixed problem
a((σ, γ), (τ , ξ)) + b((τ , ξ), (β, r, w)) = 0 ∀(τ , ξ) ∈ H,
b((σ, γ), (η, s, v)) = − gv ∀(η, s, v) ∈ Q,
Ω
where
t2
a((σ, γ), (τ , ξ)) := C −1 σ : τ + γ · ξ,
Ω κ Ω
b((τ , ξ), (η, s, v)) := η · (div τ + ξ) + s(τ12 − τ21 ) + vdiv ξ.
Ω Ω Ω
In the analysis we will utilize the following t-dependent norm for the space H
(τ , ξ) H := τ 0,Ω + div τ + ξ 0,Ω +t ξ 0,Ω + div ξ 0,Ω ,
while for the space Q, we will use
(η, s, v) Q := η 0,Ω + s 0,Ω + v 0,Ω .
Mixed FEM for Reissner-Mindlin plates. –6– B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
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7. The continuous formulation. (cont.)
Additional regularity
Proposition 1 Suppose that Ω is a convex polygon and g ∈ L2 (Ω). Then, there
a
exists a constant C , independent of t and g , such that
w 2,Ω + β 2,Ω + γ H(div ;Ω) +t γ 1,Ω + σ 1,Ω +t div σ 1,Ω + r 1,Ω ≤C g 0,Ω .
a
D. N. A RNOLD AND R. S. FALK, A uniformly accurate finite element method for the Reissner-Mindlin plate,
SIAM J. Numer. Anal., 26 (1989) 1276–1290.
Mixed FEM for Reissner-Mindlin plates. –7– B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
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8. The continuous formulation. (cont.)
Ellipticity in the kernel
V := {(τ , ξ) ∈ H : ξ + div τ = 0, τ = τ t and div ξ = 0 in Ω}.
Lemma 1 There exists C > 0, independent of t, such that
2
a((τ , ξ), (τ , ξ)) ≥ C (τ , ξ) H ∀(τ , ξ) ∈ V.
Proof. Given (τ , ξ) ∈ V , using tr(τ )2 ≤ 2(τ : τ ) ∀τ ∈ [L2 (Ω)]2×2 , we obtain
12(1 − ν) 2 t2 2
a((τ , ξ), (τ , ξ)) ≥ τ 0,Ω + ξ 0,Ω .
E κ
Since div τ + ξ 0,Ω = 0 and div ξ 0,Ω = 0, we get
2 2
a((τ , ξ), (τ , ξ)) ≥ C τ 0,Ω + div τ + ξ 0,Ω + t2 ξ 2
0,Ω + div ξ 2
0,Ω ,
2
⇒ a((τ , ξ), (τ , ξ)) ≥ C (τ , ξ) H,
Mixed FEM for Reissner-Mindlin plates. –8– B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
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9. The continuous formulation. (cont.)
Inf-Sup condition
Lemma 2 There exists C > 0, independent of t, such that
|b((τ , ξ), (η, s, v))|
sup ≥ C (η, s, v) Q ∀(η, s, v) ∈ Q.
(τ ,ξ)∈H (τ , ξ) H
(τ ,ξ)=0
Well-posedness
Theorem 2 There exists a unique ((σ, γ), (β, r, w)) ∈ H × Q solution of the
continuous mixed problem and
((σ, γ), (β, r, w)) H×Q ≤C g 0,Ω ,
where C is independent of t.
Mixed FEM for Reissner-Mindlin plates. –9– B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
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10. Galerkin formulation.
¯
• {Th }h>0 : regular family of triangulations of the polygonal region Ω.
• hT : diameter of the triangle T ∈ Th .
• h := max{hT : T ∈ Th }.
¯
• Ω= {T : T ∈ Th }.
Finite elements subspaces
γ
Hh := {ξh ∈ H(div ; Ω) : ξh |T ∈ RT0 (T ), ∀T ∈ Th },
Qw := {vh ∈ L2 (Ω) : vh |T ∈ P0 (T ), ∀T ∈ Th },
h
Mixed FEM for Reissner-Mindlin plates. – 10 – B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
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11. Galerkin formulation. (cont.)
We consider the unique polynomial bT ∈ P3 (T ) that vanishes on ∂T and is
normalized by T bT = 1.
B(Th ) := {τ ∈ H(div ; Ω) : (τi1 , τi2 )|T ∈ span{curl(bT )}, i = 1, 2, ∀T ∈ Th } ,
where curl v := (∂2 v, −∂1 v).
Hh := {τ h ∈ H(div ; Ω) : τ h |T ∈ [RT0 (T )t ]2 , ∀T ∈ Th } ⊕ B(Th ),
σ
Qβ := {ηh ∈ [L2 (Ω)]2 : ηh |T ∈ [P0 (T )]2 , ∀T ∈ Th },
h
Qr := sh ∈ H 1 (Ω) : sh |T ∈ P1 (T ), ∀T ∈ Th ,
h
Note that Hh
σ
× Qβ × Qr correspond to the PEERSa finite elements.
h h
a
D. N. A RNOLD, F. B REZZI AND J. D OUGLAS, PEERS: A new mixed finite element for the plane elasticity,
Japan J. Appl. Math., 1 (1984) 347–367.
Mixed FEM for Reissner-Mindlin plates. – 11 – B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
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12. Galerkin formulation. (cont.)
Discrete mixed problem:
Find ((σ h , γh ), (βh , rh , wh )) ∈ Hh × Qh such that
a((σ h , γh ), (τ h , ξh )) + b((τ h , ξh ), (βh , rh , wh )) = 0 ∀(τ h , ξh ) ∈ Hh ,
b((σ h , γh ), (ηh , sh , vh )) = − gvh ∀(ηh , sh , vh ) ∈ Qh .
Ω
γ
Hh := Hh × Hh ,
σ
Qh := Qβ × Qr × Qw .
h h h
Mixed FEM for Reissner-Mindlin plates. – 12 – B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
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13. Galerkin formulation. (cont.)
Discrete kernel
Vh := (τ h , ξh ) ∈ Hh : ηh · (div τ h + ξh ) + sh (τ12h − τ21h )
Ω Ω
+ vh div ξh = 0 ∀(ηh , sh , vh ) ∈ Qh .
Ω
Let (τ h , ξh )∈ Vh . Taking (0, 0, vh ) ∈ Qh and using that (div ξh )|T is a constant,
since vh |T is also a constant, we conclude that div ξh = 0 in Ω.
Now, taking (ηh , 0, 0)∈ Qh , since div τ h = 0 in Ω ∀τ h ∈ B(Th ), we have that
(div τ h )|T is a constant vector. Since div ξh = 0, we have that ξh |T is also a
constant vector. Therefore, since ηh |T is also a constant vector, we conclude that
(div τ h + ξh ) = 0 in Ω. Thus, we obtain
Vh = (τ h , ξh ) ∈ Hh : ξh + div τ h = 0, sh (τ12h − τ21h ) = 0 ∀sh ∈ Qr
h
Ω
and div ξh = 0 in Ω .
Mixed FEM for Reissner-Mindlin plates. – 13 – B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
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14. Galerkin formulation. (cont.)
Ellipticity in the discrete kernel
Lemma 3 There exists C > 0 such that
2
a((τ h , ξh ), (τ h , ξh )) ≥ C (τ h , ξh ) H ∀(τ h , ξh ) ∈ Vh ,
where the constant C is independent of h and t.
Discrete inf-Sup condition
Lemma 4 There exists C > 0, independent of h and t, such that
|b((τ h , ξh ), (ηh , sh , vh ))|
sup ≥ C (ηh , sh , vh ) Q ∀(ηh , sh , vh ) ∈ Qh .
(τ h ,ξh )∈Hh (τ h , ξh ) H
(τ h ,ξh )=0
Mixed FEM for Reissner-Mindlin plates. – 14 – B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
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15. Galerkin formulation. (cont.)
Theorem 3 There exists a unique ((σ h , γh ), (βh , rh , wh ))
∈ Hh × Qh solution of
the discrete mixed problem. Moreover, there exist C, C > 0, independent of h and t,
such that
((σ h , γh ), (βh , rh , wh )) H×Q ≤C g 0,Ω ,
and
((σ, γ), (β, r, w)) − ((σ h , γh ), (βh ,rh , wh )) H×Q
≤C inf ((σ, γ),(β, r, w)) − ((τ h , ξh ), (ηh , sh , vh )) H×Q ,
(τ h ,ξh )∈Hh
(ηh ,sh ,vh )∈Qh
where ((σ, γ), (β, r, w)) ∈ H × Q is the unique solution of the continuous mixed
problem.
Mixed FEM for Reissner-Mindlin plates. – 15 – B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
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16. Galerkin formulation. (cont.)
Rate of convergence
Theorem 4 Let ((σ, γ), (β, r, w)) ∈ H × Q and
((σ h , γh ), (βh , rh , wh )) ∈ Hh × Qh be the unique solutions of the continuous and
discrete mixed problem, respectively. If g ∈ H 1 (Ω), then,
((σ, γ), (β, r, w)) − ((σ h , γh ), (βh , rh , wh )) H×Q ≤ Ch g 1,Ω .
Mixed FEM for Reissner-Mindlin plates. – 16 – B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
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17. Numerical tests.
• Isotropic and homogeneous plate.
• Ω := (0, 1) × (0, 1).
• t = 0.001.
• E = 1, ν = 0.30 and k = 5/6.
Figure 1: Square plate: uniform meshes.
Mixed FEM for Reissner-Mindlin plates. – 17 – B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
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18. Numerical tests. (cont.)
Choosing the load g as:
E
g(x, y) = 12y(y − 1)(5x2 − 5x + 1) 2y 2 (y − 1)2
12(1 − ν 2 )
+x(x − 1)(5y 2 − 5y + 1) + 12x(x − 1)(5y 2 − 5y + 1) 2x2 (x − 1)2
+y(y − 1)(5x2 − 5x + 1) ,
so that
1 3
w(x, y) = x (x − 1)3 y 3 (y − 1)3
3
2t2
− y 3 (y − 1)3 x(x − 1)(5x2 − 5x + 1)
5(1 − ν)
+ x3 (x − 1)3 y(y − 1)(5y 2 − 5y + 1) ,
β1 (x, y) =y 3 (y − 1)3 x2 (x − 1)2 (2x − 1),
β2 (x, y) =x3 (x − 1)3 y 2 (y − 1)2 (2y − 1).
Mixed FEM for Reissner-Mindlin plates. – 18 – B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
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19. Numerical tests. (cont.)
e(σ) := σ − σ h 0,Ω , e(γ) := γ − γh t,H(div ;Ω) ,
e(β) := β − βh 0,Ω , e(r) := r − rh 0,Ω , e(w) := w − wh 0,Ω ,
log(e(·)/e′ (·))
rc(·) := −2 ,
log(N/N ′ )
where N and N ′ denote the degrees of freedom of two consecutive triangulations with
errors e and e′ .
Mixed FEM for Reissner-Mindlin plates. – 19 – B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
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20. Numerical tests. (cont.)
Table 1: Errors and experimental rates of convergence for variables σ and γ , computed
on uniform meshes.
N e(σ) rc(σ) e(γ) rc(γ)
1345 0.40270e-04 – 0.31715e-02 –
5249 0.19649e-04 1.054 0.15876e-02 1.016
20737 0.09760e-04 1.019 0.07942e-02 1.008
82433 0.04868e-04 1.008 0.03971e-02 1.004
328705 0.02431e-04 1.004 0.01986e-02 1.002
Mixed FEM for Reissner-Mindlin plates. – 20 – B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
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21. Numerical tests. (cont.)
Table 2: Errors and experimental rates of convergence for variables β , r and w, com-
puted on uniform meshes.
N e(β) rc(β) e(r) rc(r) e(w) rc(w)
1345 0.39713e-04 – 0.87462e-04 – 0.66226e-05 –
5249 0.18189e-04 1.147 0.39217e-04 1.178 0.27707e-05 1.280
20737 0.08884e-04 1.043 0.15009e-04 1.398 0.13136e-05 1.086
82433 0.04416e-04 1.013 0.05491e-04 1.457 0.06478e-05 1.025
328705 0.02205e-04 1.004 0.01991e-04 1.466 0.03228e-05 1.007
Mixed FEM for Reissner-Mindlin plates. – 21 – B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
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22. Numerical tests. (cont.)
10
σ ♦
1 γ +
β
0.1 ⋆ r ×
⋆ w △
⋆
0.01 ⋆ h⋆ ⋆
+
0.001 +
+
e +
+
0.0001 ×
♦ ×
♦ ×
1e-05 △ ♦ ×
△ ♦
△ ♦
×
1e-06 △
△
1e-07
1e-08
1000 10000 100000 1e+06
degrees of freedom N
Mixed FEM for Reissner-Mindlin plates. – 22 – B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
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23. Numerical tests. (cont.)
Figure 2: Approximate transverse displacement (left) and first component of the rotation
vector (right).
Mixed FEM for Reissner-Mindlin plates. – 23 – B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
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24. Numerical tests. (cont.)
Figure 3: Approximate shear vector: first component (left) and second component
(right).
Mixed FEM for Reissner-Mindlin plates. – 24 – B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
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25. Numerical tests. (cont.)
Figure 4: Approximate bending moment: σ 11h (left) and σ 12h (right).
Mixed FEM for Reissner-Mindlin plates. – 25 – B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
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26. Numerical tests. (cont.)
Figure 5: Approximate bending moment: σ 21h (left) and σ 22h (right).
Mixed FEM for Reissner-Mindlin plates. – 26 – B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
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27. Many thanks for your attention.
Mixed FEM for Reissner-Mindlin plates. – 27 – B EIR AO DA V EIGA , M ORA , R ODR´GUEZ
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