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: A Bakhtinian homecoming: Potentials and pitfalls for dialogic pedagogy in a contemporary world

authors: White, Elizabeth Jayne

subjects: Bakhtin

abstract: Bakhtin allows us to approach acts as the embryo or seeds of becoming – as existential -  rather than the products of objectified being based on valued knowledge- in doing so he provides a rich research landscape for investigating metaphoricity.

text: A Bakhtinian homecoming: Potentials and pitfalls for dialogic pedagogy in a contemporary world Bakhtin allows us to approach acts as the embryo or seeds of becoming – as existential -  rather than the products of objectified being based on valued knowledge- in doing so he provides a rich research landscape for investigating metaphoricity.

year: 2011

title: Whanaungatanga: an illustration of the importance of cultural context

authors: McNatty, William

subjects: Te reo Māori

abstract: The word 'whanaungatanga' is often used by people who lack an intimate understanding of the philosophical base in Māori culture out of which it emerges. Thus, it is sometimes used simply to describe the camaraderie that can be associated with team sports or with reference to the 'glue' that connects people to one another socially. This is not necessarily a problem in itself. After all, change and development in the lexicon is a characteristic of living languages. However, real problems can emerge where words become divorced from their cultural roots to such an extent that two speakers using the same word can completely fail to understand one another. It is argued here that this is what can happen when Māori words, such as 'whanaungatanga', are used in the context of modelling in the social sciences, particularly where that modelling is conducted in the context of Western social and cultural values. In particular, the essential spiritual dimension of whanaungatanga may be overlooked or misunderstood.

text: Whanaungatanga: an illustration of the importance of cultural context The word 'whanaungatanga' is often used by people who lack an intimate understanding of the philosophical base in Māori culture out of which it emerges. Thus, it is sometimes used simply to describe the camaraderie that can be associated with team sports or with reference to the 'glue' that connects people to one another socially. This is not necessarily a problem in itself. After all, change and development in the lexicon is a characteristic of living languages. However, real problems can emerge where words become divorced from their cultural roots to such an extent that two speakers using the same word can completely fail to understand one another. It is argued here that this is what can happen when Māori words, such as 'whanaungatanga', are used in the context of modelling in the social sciences, particularly where that modelling is conducted in the context of Western social and cultural values. In particular, the essential spiritual dimension of whanaungatanga may be overlooked or misunderstood.

year: 2002-02