title: The use of positive and negative penalty functions in solving constrained optimization problems and partial differential equations

authors: Monterrubio Salazar, Luis Emilio

subjects: Rayleigh-Ritz Method

abstract: The Rayleigh-Ritz Method together with the Penalty Function Method is used to investigate the use of different types of penalty parameters. The use of artificial springs as penalty parameters is a very well established procedure to model constraints in the Rayleigh-Ritz Method, the Finite Element Method and other numerical methods. Historically, large positive values were used to define the stiffness coefficient of artificial springs, until recent publications demonstrated that it is possible to use negative values to define the stiffness coefficients of the springs. Furthermore, recent publications show that constraints can be enforced using positive and negative mass or inertia in vibration problems and in a more generic sense using eigenpenalty parameters which are penalty parameters in the matrix associated with the eigenvalue. Before the commencement of this thesis, solutions using artificial inertia were published only for beams and simple spring-mass systems.
In this thesis the use of all possible types of penalty parameters are investigated in vibration problems of Euler-Bernoulli beams, thin plates and shallow shells and in elastic stability analysis of Euler-Bernoulli beams, including penalty parameters associated with the geometrical stiffness matrix. The study includes the use of penalty parameters for both enforcing support boundary conditions and continuity conditions along structural joints.
This investigation started with the selection of the set of admissible functions that would: (a) allow modelling of beams, plates and shells in completely free boundary conditions; (b) not present any limitation in the number of functions that can be used in the solution. This gives the possibility to converge to the constraint solution and to model any type of boundary conditions.
The procedure proposed in this work combines several advantages: accuracy of the results, relative fast convergence, simplicity of the set of admissible functions and flexibility to define boundary conditions. While there are other procedures that may give better accuracy for specific cases, the proposed method is more widely applicable.
The procedure used in this work also includes a way to check for round-off errors and ill-conditioning in the results; as well as a way to bracket the  exact  solution with upper and lower-bound results.

text: The use of positive and negative penalty functions in solving constrained optimization problems and partial differential equations The Rayleigh-Ritz Method together with the Penalty Function Method is used to investigate the use of different types of penalty parameters. The use of artificial springs as penalty parameters is a very well established procedure to model constraints in the Rayleigh-Ritz Method, the Finite Element Method and other numerical methods. Historically, large positive values were used to define the stiffness coefficient of artificial springs, until recent publications demonstrated that it is possible to use negative values to define the stiffness coefficients of the springs. Furthermore, recent publications show that constraints can be enforced using positive and negative mass or inertia in vibration problems and in a more generic sense using eigenpenalty parameters which are penalty parameters in the matrix associated with the eigenvalue. Before the commencement of this thesis, solutions using artificial inertia were published only for beams and simple spring-mass systems.
In this thesis the use of all possible types of penalty parameters are investigated in vibration problems of Euler-Bernoulli beams, thin plates and shallow shells and in elastic stability analysis of Euler-Bernoulli beams, including penalty parameters associated with the geometrical stiffness matrix. The study includes the use of penalty parameters for both enforcing support boundary conditions and continuity conditions along structural joints.
This investigation started with the selection of the set of admissible functions that would: (a) allow modelling of beams, plates and shells in completely free boundary conditions; (b) not present any limitation in the number of functions that can be used in the solution. This gives the possibility to converge to the constraint solution and to model any type of boundary conditions.
The procedure proposed in this work combines several advantages: accuracy of the results, relative fast convergence, simplicity of the set of admissible functions and flexibility to define boundary conditions. While there are other procedures that may give better accuracy for specific cases, the proposed method is more widely applicable.
The procedure used in this work also includes a way to check for round-off errors and ill-conditioning in the results; as well as a way to bracket the  exact  solution with upper and lower-bound results.

year: 2009

title: Characteristics of temporal fluctuations in the hyperpolarized state of the cortical slow oscillation

authors: Wilson, Marcus T.

subjects: bioelectric potentials

abstract: We present evidence for the hypothesis that transitions between the low- and high-firing states of the cortical slow oscillation correspond to neuronal phase transitions. By analyzing intracellular recordings of the membrane potential during the cortical slow oscillation in rats, we quantify the temporal fluctuations in power and the frequency centroid of the power spectrum in the period of time before “down” to “up” transitions. By taking appropriate averages over such events, we present these statistics as a function of time before transition. The results demonstrate an increase in fluctuation power and time scale broadly consistent with the slowing of systems close to phase transitions. The analysis is complicated and limited by the difficulty in identifying when transitions begin, and removing dc trends in membrane potential.

text: Characteristics of temporal fluctuations in the hyperpolarized state of the cortical slow oscillation We present evidence for the hypothesis that transitions between the low- and high-firing states of the cortical slow oscillation correspond to neuronal phase transitions. By analyzing intracellular recordings of the membrane potential during the cortical slow oscillation in rats, we quantify the temporal fluctuations in power and the frequency centroid of the power spectrum in the period of time before “down” to “up” transitions. By taking appropriate averages over such events, we present these statistics as a function of time before transition. The results demonstrate an increase in fluctuation power and time scale broadly consistent with the slowing of systems close to phase transitions. The analysis is complicated and limited by the difficulty in identifying when transitions begin, and removing dc trends in membrane potential.

year: 2008

title: Te Kōtihitihi: Ngā Tuhinga Reo Māori, Putanga 2

authors: Ngāpō, Korohere

abstract: Koinei te katoa o te Putanga 2 o te puka hautaka, Te Kōtihitihi.

text: Te Kōtihitihi: Ngā Tuhinga Reo Māori, Putanga 2 Koinei te katoa o te Putanga 2 o te puka hautaka, Te Kōtihitihi.

year: 2012-05