Scattering on the Timoshenko Beam. Direct and Inverse Problems in the Time Domain

This thesis concerns the use of time domain methods for treating direct and inverse scattering problems for flexural waves on beams. The Timoshenko equation is used throughout to describe the behaviour of such waves. In the theoretical part of the work, the beam is considered as inhomogeneous as allowed by elementary beam theory, excluding side bending. Furthermore, the possibility of the beam to be resting on a viscoelastic foundation is allowed for. The damping influence of the foundation is characterised by constitutive relations involving the past history of beam rotation and deflection. Together, the suspension and the inhomogeneity of the beam constitute the region of scattering. The key to the direct and inverse problems lies in the scattering operators of this region. However, the concept of wave splitting is equally important. Basically this is a transformation of the equations governing the unrestrained and homogeneous beam, into equations that describe split wave fields propagating independently in definite directions. The wave splitting is necessary in order to decompose the flexural wave field at the boundary of incidence into its incoming and reflected parts. Two different approaches to obtaining the scattering operators for the Timoshenko beam are used. First, there are the imbedding reflection and transmission operators that map an incident field, impinging on the boundary of the scattering region, to the field reflected at the same boundary, and the transmitted field at the back end of the region. This method essentially is a homotopy approach with the idea of imbedding the original problem in a family of related problems. The homotopy approach is used to imbed the scattering problem of the full region in a one-parameter family of sub-region scattering problems. Second, there is the wave propagator formalism which is more general in that the corresponding scattering operators, the wave propagators, map the incident field onto the split fields at any internal position of the scattering region. This formalism contains the imbedding method as a special case. The scattering operators mentioned above have explicit representations in terms of integral kernels that satisfy matrix-valued integro-differential equations. These equations are derived and numerical methods for solving these equations are presented. Numerical solutions are obtained for two types of scatterers: a homogeneous beam on a viscoelastic foundation, modelled by exponential memory functions, and a non-uniform unrestrained beam. Moreover, the imbedding reflection equation is used to set up an explicit inverse algorithm in order to determine the variation of a non-uniform cross section from knowledge of reflection data. Examples of simulated noisy reconstructions are given for both circular and rectangular cross-sections

Scattering on the Timoshenko Beam. Direct and Inverse Problems in the Time Domain

This thesis concerns the use of time domain methods for treating direct and inverse scattering problems for flexural waves on beams. The Timoshenko equation is used throughout to describe the behaviour of such waves. In the theoretical part of the work, the beam is considered as inhomogeneous as allowed by elementary beam theory, excluding side bending. Furthermore, the possibility of the beam to be resting on a viscoelastic foundation is allowed for. The damping influence of the foundation is characterised by constitutive relations involving the past history of beam rotation and deflection. Together, the suspension and the inhomogeneity of the beam constitute the region of scattering. The key to the direct and inverse problems lies in the scattering operators of this region. However, the concept of wave splitting is equally important. Basically this is a transformation of the equations governing the unrestrained and homogeneous beam, into equations that describe split wave fields propagating independently in definite directions. The wave splitting is necessary in order to decompose the flexural wave field at the boundary of incidence into its incoming and reflected parts. Two different approaches to obtaining the scattering operators for the Timoshenko beam are used. First, there are the imbedding reflection and transmission operators that map an incident field, impinging on the boundary of the scattering region, to the field reflected at the same boundary, and the transmitted field at the back end of the region. This method essentially is a homotopy approach with the idea of imbedding the original problem in a family of related problems. The homotopy approach is used to imbed the scattering problem of the full region in a one-parameter family of sub-region scattering problems. Second, there is the wave propagator formalism which is more general in that the corresponding scattering operators, the wave propagators, map the incident field onto the split fields at any internal position of the scattering region. This formalism contains the imbedding method as a special case. The scattering operators mentioned above have explicit representations in terms of integral kernels that satisfy matrix-valued integro-differential equations. These equations are derived and numerical methods for solving these equations are presented. Numerical solutions are obtained for two types of scatterers: a homogeneous beam on a viscoelastic foundation, modelled by exponential memory functions, and a non-uniform unrestrained beam. Moreover, the imbedding reflection equation is used to set up an explicit inverse algorithm in order to determine the variation of a non-uniform cross section from knowledge of reflection data. Examples of simulated noisy reconstructions are given for both circular and rectangular cross-sections

Wave propagators for the Timoshenko beam

The propagation and scattering of waves on the Timoshenko beam are investigated by using the method of wave propagators. This method is more general than the scattering operators connected to the imbedding and Green function approaches; the wave propagators map the incoming field at an internal position onto the scattering fields at any other internal position of the scattering region. This formalism contains the imbedding method and Green function approach as special cases. Equations for the propagator kernels are derived, as are the conditions for their discontinuities. Symmetry requirements on certain coupling matrices originating from the wave splitting are considered. They are illustrated by two specific examples. The first being an unrestrained beam with a varying cross-section and the other a homogeneous, viscoelastically restrained beam. A numerical algorithm for solving the equations for the propagator kernels is described. The algorithm is tested for the case of a viscoelastically restrained, homogeneous beam. In a limit these results agree with the ones obtained for the reflection kernel by a previously developed algorithm for the imbedding reflection equation

The imbedding equations for the Timoshenko beam

Wave reflection in a Timoshenko beam is treated, using wave splitting and the imbedding technique. The beam is assumed to be inhomogeneous and restrained by a viscoelastic suspension. The viscoelasticity is characterized by constitutive relations that involve the past history of deflection and rotation of the beam through memory functions of the suspension. By applying wave splitting, the propagating fields are decomposed into left- and right-moving parts. An integral representation of the split fields in impulse responses is presented. This representation gives the reflected and transmitted fields as convolutions of the incident field with the reflection and transmission kernels, respectively. The kernels are independent of the incident field and depend only on the material properties. Invariant imbedding is used to obtain equations for these kernels. In general, the kernels contain discontinuities for which transport equations are derived and solved. Some numerical solutions are presented for the reflection by a homogeneous beam suspended on two separated, semi-infinite layers of continuously distributed, viscoelastically damped, local acting springs

URBAN MODELLING AND VISUALIZATION TOOLS FOR URBAN TRANSPORTATION SYSTEMS – EXAMPLES FROM TWO LIVING-LAB PROJECTS IN GOTHENBURG.

Abstract:Purpose: Within transport projects there is a growing demand for urban modelling and advanced visualization methods. This paper reflects upon visualizations techniques used in two transdisciplinary projects studying implementation of transport solutions in Gothenburg. Involvement of many stakeholders and efficient dialogue tools were essential to support communication in the transdisciplinary environment. Sendsmart and Go:Smart projects (2012-2014) aimed at developing and testing innovative sustainable solutions for urban transportation with a focus on freight (Sendsmart) and passenger transport (Go:Smart). They were developed as an important municipal attempt to create better conditions for sustainable urban travel in the city. Research Approach: Both projects turned out to become a living laboratory for visualization implementation and engaged groups of key stakeholders from the academia, industry, city of Gothenburg, and the regional and national organizations. These practice-oriented projects were focused on development of new solutions and testing them in reality. This study presents reflections from a research-by-design process and available rich, documented material from the projects (meeting notes, workshop notes, monthly reports, films). Even if, SendSmart nor GoSmart were not part of municipal planning process, they were focused on early implementation phase of new approaches in the city transport planning. Integration of users was essential and the user perspective was the only one brought into research discourse.Findings and Originality: In both projects methods and tools were developed in forms of demo visualizations and films, simulation models - scenario development and evaluation (decision support systems: Urban Strategy combined with Visum), image supported discussions (Urbania) maps and 2D visualizations as a basis for discussion. These tools are perceived as extremely helpful to support communication in the complex environments and were very useful as an input to the workshops. However, an iterative procedure would have been needed to further let the participants’ opinions and suggestions lead to new visualized concepts. A need to simulate both in macro and micro scale was recognized. Challenges to further deal with are lack of detailed data for traffic simulation in advanced models, problems with different source data aggregation and a high demand for specifically qualified expertise in building simulation models. It is beneficial to put efforts into developing an integrated model for freight and passenger transport within transdisciplinary projects. Research Impact: This paper underlines the necessity for a critical collaborative exchange and research needs to be fostered and disseminated in order to enhance and promote the usable knowledge and application of visualization methods and technologies. Their potential in addressing critical transportation issues of today, as well as promoting innovative approaches to meet society’s transportation needs of the future often requires a discussion within a broader, multidisciplinary context of technology development in the areas of simulation and modellingPractical impact: Paper addresses the importance of using visualization for communication in transportation projects

URBAN MODELLING AND VISUALIZATION TOOLS FOR URBAN TRANSPORTATION SYSTEMS – EXAMPLES FROM TWO LIVING-LAB PROJECTS IN GOTHENBURG.

Abstract:Purpose: Within transport projects there is a growing demand for urban modelling and advanced visualization methods. This paper reflects upon visualizations techniques used in two transdisciplinary projects studying implementation of transport solutions in Gothenburg. Involvement of many stakeholders and efficient dialogue tools were essential to support communication in the transdisciplinary environment. Sendsmart and Go:Smart projects (2012-2014) aimed at developing and testing innovative sustainable solutions for urban transportation with a focus on freight (Sendsmart) and passenger transport (Go:Smart). They were developed as an important municipal attempt to create better conditions for sustainable urban travel in the city. Research Approach: Both projects turned out to become a living laboratory for visualization implementation and engaged groups of key stakeholders from the academia, industry, city of Gothenburg, and the regional and national organizations. These practice-oriented projects were focused on development of new solutions and testing them in reality. This study presents reflections from a research-by-design process and available rich, documented material from the projects (meeting notes, workshop notes, monthly reports, films). Even if, SendSmart nor GoSmart were not part of municipal planning process, they were focused on early implementation phase of new approaches in the city transport planning. Integration of users was essential and the user perspective was the only one brought into research discourse.Findings and Originality: In both projects methods and tools were developed in forms of demo visualizations and films, simulation models - scenario development and evaluation (decision support systems: Urban Strategy combined with Visum), image supported discussions (Urbania) maps and 2D visualizations as a basis for discussion. These tools are perceived as extremely helpful to support communication in the complex environments and were very useful as an input to the workshops. However, an iterative procedure would have been needed to further let the participants’ opinions and suggestions lead to new visualized concepts. A need to simulate both in macro and micro scale was recognized. Challenges to further deal with are lack of detailed data for traffic simulation in advanced models, problems with different source data aggregation and a high demand for specifically qualified expertise in building simulation models. It is beneficial to put efforts into developing an integrated model for freight and passenger transport within transdisciplinary projects. Research Impact: This paper underlines the necessity for a critical collaborative exchange and research needs to be fostered and disseminated in order to enhance and promote the usable knowledge and application of visualization methods and technologies. Their potential in addressing critical transportation issues of today, as well as promoting innovative approaches to meet society’s transportation needs of the future often requires a discussion within a broader, multidisciplinary context of technology development in the areas of simulation and modellingPractical impact: Paper addresses the importance of using visualization for communication in transportation projects

MEMS Compact Modeling Meets Model Order Reduction: Examples of the

Modeling and simulation of the behavior of a system consisting of many single devices is an essential requirement for the reduction of design cycles in the development of microsystem applications. Analytic solutions for the describing partial differential equations of each component are only available for simple geometries. For complex geometries, either approximations or numerical methods can be used. However, the numerical treatment of the PDEs of thousands of interconnected single devices with each exhibiting a complex behavior is almost impossible without reduction of the order of unknowns to a lower-dimensional system. We present a fully automatic method to generate a compact model of second-order linear systems based on the Arnoldi process, and provide an example of successfull model order reduction to a gyroscope

Visualizing environmental data for pedestrian comfort analysis in urban planning processes

Digital tools are being developed for involving stakeholders in urban planning and transformation processes. One challenge is how to visualize and act upon all parameters that are relevant for dealing with complex plan-ning problems, such as environmental factors. Dialogue tools involving vis-ualization can bridge the distance between planners and citizens. This paper focuses on the problem of representing invisible environmental parameters affecting the urban climate such as wind, solar radiation, air pollution and noise, in a city model. The aim of the paper is to discuss challenges for rep-resenting and communicating environmental data in city models. In this pa-per, we have accounted for literature studies; our own conceptual modelling and prototype studies in three projects; as well as a survey with 16 urban planners. We conclude with defining design criteria for dialogue tools cre-ating a comprehensible base for communication in urban transformation processes