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Nonlinear Eigenvalue Problems General Theory, Results and Methods Nonlinear Eigenvalue Problems General Theory, Results and Methods Heinrich Voss [email protected] Hamburg University of Technology Institute of Mathematics TUHH Heinrich Voss Isfahan, July, 2016 1 / 72 On the one hand: Parameter dependent nonlinear (with respect to the state variable) operator equations T (λ, u) = 0 discussing — positivity of solutions — multiplicity of solution — dependence of solutions on the parameter; bifurcation — (change of ) stability of solutions Nonlinear eigenvalue problem The term nonlinear eigenvalue problem is not used in a unique way in the literature TUHH Heinrich Voss Isfahan, July, 2016 2 / 72 Nonlinear eigenvalue problem The term nonlinear eigenvalue problem is not used in a unique way in the literature On the one hand: Parameter dependent nonlinear (with respect to the state variable) operator equations T (λ, u) = 0 discussing — positivity of solutions — multiplicity of solution — dependence of solutions on the parameter; bifurcation — (change of ) stability of solutions TUHH Heinrich Voss Isfahan, July, 2016 2 / 72 Find λ 2 D and x 6= 0 such that T (λ)x = 0: Then λ is called an eigenvalue of T (·), and x a corresponding (right) eigenvector. In this presentation For D ⊂ C let T (λ), λ 2 D be a family of linear operators on a Hilbert space H (more generally a family of closed linear operators on a Banach space). TUHH Heinrich Voss Isfahan, July, 2016 3 / 72 In this presentation For D ⊂ C let T (λ), λ 2 D be a family of linear operators on a Hilbert space H (more generally a family of closed linear operators on a Banach space). Find λ 2 D and x 6= 0 such that T (λ)x = 0: Then λ is called an eigenvalue of T (·), and x a corresponding (right) eigenvector. TUHH Heinrich Voss Isfahan, July, 2016 3 / 72 Outline 1 Examples 2 Basic Properties 3 Rayleigh Functional 4 Methods for dense nonlinear eigenvalue problems Linearization Methods based on characteristic equation Newton’s method 5 Invariant pairs TUHH Heinrich Voss Isfahan, July, 2016 4 / 72 Examples Outline 1 Examples 2 Basic Properties 3 Rayleigh Functional 4 Methods for dense nonlinear eigenvalue problems Linearization Methods based on characteristic equation Newton’s method 5 Invariant pairs TUHH Heinrich Voss Isfahan, July, 2016 5 / 72 i! t Suppose that the system is exited by a time harmonic force f (t) = f0e 0 . If C = αK + βM (modal damping) and (xj ; µj ), j = 1;:::; n denotes an T orthonormal eigensystem of Kx = µMx (i.e. xk Mxj = δjk ) then the periodic response of the system is n T X xj f0 q(t) = ei!0t x : µ − !2 + i! (αµ + β) j j=1 j 0 0 j Usually, one gets good approximations taking into account only a small number of eigenvalues in the vicinity of i!0 (truncated mode superposition). Examples Example 1: Vibration of structures Equations of motion arising in dynamic analysis of structures (with a finite number of degrees of freedom) are Mq¨(t) + Cq_ (t) + Kq(t) = f (t) where q are the Lagrangean coordinates, M is the mass matrix, K the stiffness matrix, C the viscous damping matrix, and f an external force vector. TUHH Heinrich Voss Isfahan, July, 2016 6 / 72 If C = αK + βM (modal damping) and (xj ; µj ), j = 1;:::; n denotes an T orthonormal eigensystem of Kx = µMx (i.e. xk Mxj = δjk ) then the periodic response of the system is n T X xj f0 q(t) = ei!0t x : µ − !2 + i! (αµ + β) j j=1 j 0 0 j Usually, one gets good approximations taking into account only a small number of eigenvalues in the vicinity of i!0 (truncated mode superposition). Examples Example 1: Vibration of structures Equations of motion arising in dynamic analysis of structures (with a finite number of degrees of freedom) are Mq¨(t) + Cq_ (t) + Kq(t) = f (t) where q are the Lagrangean coordinates, M is the mass matrix, K the stiffness matrix, C the viscous damping matrix, and f an external force vector. i! t Suppose that the system is exited by a time harmonic force f (t) = f0e 0 . TUHH Heinrich Voss Isfahan, July, 2016 6 / 72 Examples Example 1: Vibration of structures Equations of motion arising in dynamic analysis of structures (with a finite number of degrees of freedom) are Mq¨(t) + Cq_ (t) + Kq(t) = f (t) where q are the Lagrangean coordinates, M is the mass matrix, K the stiffness matrix, C the viscous damping matrix, and f an external force vector. i! t Suppose that the system is exited by a time harmonic force f (t) = f0e 0 . If C = αK + βM (modal damping) and (xj ; µj ), j = 1;:::; n denotes an T orthonormal eigensystem of Kx = µMx (i.e. xk Mxj = δjk ) then the periodic response of the system is n T X xj f0 q(t) = ei!0t x : µ − !2 + i! (αµ + β) j j=1 j 0 0 j Usually, one gets good approximations taking into account only a small number of eigenvalues in the vicinity of i!0 (truncated mode superposition). TUHH Heinrich Voss Isfahan, July, 2016 6 / 72 Discretizing by finite elements yields (cf. Hager & Wiberg 2000) K 2 X 1 T (!)x := ! M + K − ∆Kj x = 0 1 + bj ! j=1 where M is the consistent mass matrix, K is the stiffness matrix with the instantaneous elastic material parameters used in Hooke’s law, and ∆Kj collects the contributions of damping from elements with relaxation parameter bj . Examples Example 2: Viscoelastic model of damping Using a viscoelastic constitutive relation to describe the material behavior in the equations of motion yields a rational eigenvalue problem in the case of free vibrations. TUHH Heinrich Voss Isfahan, July, 2016 7 / 72 Examples Example 2: Viscoelastic model of damping Using a viscoelastic constitutive relation to describe the material behavior in the equations of motion yields a rational eigenvalue problem in the case of free vibrations. Discretizing by finite elements yields (cf. Hager & Wiberg 2000) K 2 X 1 T (!)x := ! M + K − ∆Kj x = 0 1 + bj ! j=1 where M is the consistent mass matrix, K is the stiffness matrix with the instantaneous elastic material parameters used in Hooke’s law, and ∆Kj collects the contributions of damping from elements with relaxation parameter bj . TUHH Heinrich Voss Isfahan, July, 2016 7 / 72 In such nanostructures, the free carriers are confined to a small region of space by potential barriers, and if the size of this region is less than the electron wavelength, the electronic states become quantized at discrete energy levels. The ultimate limit of low dimensional structures is the quantum dot, in which the carriers are confined in all three directions, thus reducing the degrees of freedom to zero. Therefore, a quantum dot can be thought of as an artificial atom. Examples Example 3: Electronic structure of quantum dots Semiconductor nanostructures have attracted tremendous interest in the past few years because of their special physical properties and their potential for applications in micro– and optoelectronic devices. TUHH Heinrich Voss Isfahan, July, 2016 8 / 72 The ultimate limit of low dimensional structures is the quantum dot, in which the carriers are confined in all three directions, thus reducing the degrees of freedom to zero. Therefore, a quantum dot can be thought of as an artificial atom. Examples Example 3: Electronic structure of quantum dots Semiconductor nanostructures have attracted tremendous interest in the past few years because of their special physical properties and their potential for applications in micro– and optoelectronic devices. In such nanostructures, the free carriers are confined to a small region of space by potential barriers, and if the size of this region is less than the electron wavelength, the electronic states become quantized at discrete energy levels. TUHH Heinrich Voss Isfahan, July, 2016 8 / 72 Examples Example 3: Electronic structure of quantum dots Semiconductor nanostructures have attracted tremendous interest in the past few years because of their special physical properties and their potential for applications in micro– and optoelectronic devices. In such nanostructures, the free carriers are confined to a small region of space by potential barriers, and if the size of this region is less than the electron wavelength, the electronic states become quantized at discrete energy levels. The ultimate limit of low dimensional structures is the quantum dot, in which the carriers are confined in all three directions, thus reducing the degrees of freedom to zero. Therefore, a quantum dot can be thought of as an artificial atom. TUHH Heinrich Voss Isfahan, July, 2016 8 / 72 Examples Example 3 ct. Electron Spectroscopy Group, Fritz-Haber-Institute, Berlin TUHH Heinrich Voss Isfahan, July, 2016 9 / 72 Boundary and interface conditions Φ = 0 on outer boundary of matrix Ωm 1 @Φ 1 @Φ BenDaniel–Duke condition = on interface mm @n mq @n @Ωm @Ωq Details in Talk tomorrow m and V are discontinous across the heterojunction. Examples Example 3 ct. Governing equation: Schrodinger¨ equation 2 −∇ · ~ rΦ + V (x)Φ = EΦ; x 2 Ω [ Ω 2m(x; E) q m where ~ is the reduced Planck constant, m(x; E) is the electron effective mass depending on the energy state E, and V (x) is the confinement potential. TUHH Heinrich Voss Isfahan, July, 2016 10 / 72 Boundary and interface conditions Φ = 0 on outer boundary of matrix Ωm 1 @Φ 1 @Φ BenDaniel–Duke condition = on interface mm @n mq @n @Ωm @Ωq Details in Talk tomorrow Examples Example 3 ct. Governing equation: Schrodinger¨ equation 2 −∇ · ~ rΦ + V (x)Φ = EΦ; x 2 Ω [ Ω 2m(x; E) q m where ~ is the reduced Planck constant, m(x; E) is the electron effective mass depending on the energy state E, and V (x) is the confinement potential.
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