A finite element method (FEM) formulation for the detection of unknown steady boundary conditions in heat conduction and linear elasticity and thermoelasticity continuum problems is presented. The present FEM formulation is capable of determining displacements, surface stresses, temperatures, and heat fluxes on the boundaries where such quantities are unknown or inaccessible, provided such quantities are sufficiently over-specified on other boundaries. Details of the discretization, linear system solution techniques, and sample results for 2-D problems are presented.
{"title":"Simultaneous Determination of Temperatures, Heat Fluxes, Deformations, and Tractions on Inaccessible Boundaries","authors":"B. Dennis, G. Dulikravich","doi":"10.1115/imece1998-0215","DOIUrl":"https://doi.org/10.1115/imece1998-0215","url":null,"abstract":"\u0000 A finite element method (FEM) formulation for the detection of unknown steady boundary conditions in heat conduction and linear elasticity and thermoelasticity continuum problems is presented. The present FEM formulation is capable of determining displacements, surface stresses, temperatures, and heat fluxes on the boundaries where such quantities are unknown or inaccessible, provided such quantities are sufficiently over-specified on other boundaries. Details of the discretization, linear system solution techniques, and sample results for 2-D problems are presented.","PeriodicalId":331326,"journal":{"name":"Computational Methods for Solution of Inverse Problems in Mechanics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1998-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121621456","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper presents a method for identifying a set of spatial matrices, which are the coefficient matrices of equations of motion for mechanical structures in the physical domain. The input data for the method are a set of frequency response functions measured experimentally within a limited frequency range of interest and the coordinate data of measurement points. This is a practical engineering inverse problem. The definition of the inverse problem and the method developed are presented in this paper. In addition, a simple example is shown to demonstrate its practical validity and usefulness.
{"title":"Experimental Spatial Matrix Identification as a Practical Inverse Problem in Mechanics","authors":"M. Okuma, T. Oho","doi":"10.1115/imece1998-0222","DOIUrl":"https://doi.org/10.1115/imece1998-0222","url":null,"abstract":"\u0000 This paper presents a method for identifying a set of spatial matrices, which are the coefficient matrices of equations of motion for mechanical structures in the physical domain. The input data for the method are a set of frequency response functions measured experimentally within a limited frequency range of interest and the coordinate data of measurement points. This is a practical engineering inverse problem. The definition of the inverse problem and the method developed are presented in this paper. In addition, a simple example is shown to demonstrate its practical validity and usefulness.","PeriodicalId":331326,"journal":{"name":"Computational Methods for Solution of Inverse Problems in Mechanics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1998-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125811496","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In a recent series of papers we proposed a new class of methods, the generalized eigensystem (GES) methods, for solving the inverse problem of electrocardiography. In this paper, we compare zero, first, and second order regularized GES methods to zero, first, and second order Tikhonov methods. Results from higher order regularization depend heavily on the exact form of the regularization operator, and operators generated by finite element techniques give the most accurate and consistent results. The GES techniques always produce smaller average relative errors than the Tikhonov techniques, but as the regularization order increases the difference in average relative errors between the two techniques becomes less pronounced.
{"title":"Regularization Matrices for Inverse Electrocardiography","authors":"L. Olson, R. Throne","doi":"10.1115/imece1998-0223","DOIUrl":"https://doi.org/10.1115/imece1998-0223","url":null,"abstract":"\u0000 In a recent series of papers we proposed a new class of methods, the generalized eigensystem (GES) methods, for solving the inverse problem of electrocardiography. In this paper, we compare zero, first, and second order regularized GES methods to zero, first, and second order Tikhonov methods.\u0000 Results from higher order regularization depend heavily on the exact form of the regularization operator, and operators generated by finite element techniques give the most accurate and consistent results. The GES techniques always produce smaller average relative errors than the Tikhonov techniques, but as the regularization order increases the difference in average relative errors between the two techniques becomes less pronounced.","PeriodicalId":331326,"journal":{"name":"Computational Methods for Solution of Inverse Problems in Mechanics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1998-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114316422","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A methodology for solving inverse problems in Micro-electromechanical (MEM) systems is proposed in this paper. Design of variable shape electrostatic comb drives (shape motors), in order to obtain desired force profiles, is presented as an application of the general methodology. This analysis includes simulation, sensitivity analysis and optimization. A comb drive is one of the most important microactuators in MEM systems. In a standard comb drive, the capacitance varies linearly with displacement, resulting in an electrostatic driving force which is independent of the position of the moving fingers (relative to the fixed ones) except at the ends of the range of travel. It is of interest in some applications to have force profiles such as linear, quadratic or cubic. Such shaped comb drives could be useful, for example, for electrostatic tuning or to get actuators with longer ranges of travel than those of standard comb drives. The present paper addresses the issues of simulation, sensitivity analysis, and then design (inverse problem) of comb drives with variable height profiles. Three-dimensional simulations of the exterior electrostatic field, and the resultant forces on the comb drive, are carried out with the exterior, indirect, boundary element method. Following direct simulation, sensitivity analysis is carried out by the direct differentiation approach. The variable of interest is the driving force while the design variables are parameters that determine the shape of the moving fingers. Next, an inverse problem is posed as follows: determine the height profile of the moving fingers such that the driving force is a desired function of the displacement of the comb drive. Comb drives of appropriate shapes, that produce desired force profiles, are obtained by this approach. Numerical results are given for shape motors that produce linear or cubic force profiles as functions of travel. The optimization code “E04UCF”, from the NAG package, is used for this phase of the work.
{"title":"Optimal Shape Design of Three Dimensional MEMS With Applications to Electrostatic Comb Drives","authors":"W. Ye, S. Mukherjee","doi":"10.1115/imece1998-0217","DOIUrl":"https://doi.org/10.1115/imece1998-0217","url":null,"abstract":"\u0000 A methodology for solving inverse problems in Micro-electromechanical (MEM) systems is proposed in this paper. Design of variable shape electrostatic comb drives (shape motors), in order to obtain desired force profiles, is presented as an application of the general methodology. This analysis includes simulation, sensitivity analysis and optimization.\u0000 A comb drive is one of the most important microactuators in MEM systems. In a standard comb drive, the capacitance varies linearly with displacement, resulting in an electrostatic driving force which is independent of the position of the moving fingers (relative to the fixed ones) except at the ends of the range of travel. It is of interest in some applications to have force profiles such as linear, quadratic or cubic. Such shaped comb drives could be useful, for example, for electrostatic tuning or to get actuators with longer ranges of travel than those of standard comb drives.\u0000 The present paper addresses the issues of simulation, sensitivity analysis, and then design (inverse problem) of comb drives with variable height profiles. Three-dimensional simulations of the exterior electrostatic field, and the resultant forces on the comb drive, are carried out with the exterior, indirect, boundary element method. Following direct simulation, sensitivity analysis is carried out by the direct differentiation approach. The variable of interest is the driving force while the design variables are parameters that determine the shape of the moving fingers. Next, an inverse problem is posed as follows: determine the height profile of the moving fingers such that the driving force is a desired function of the displacement of the comb drive. Comb drives of appropriate shapes, that produce desired force profiles, are obtained by this approach. Numerical results are given for shape motors that produce linear or cubic force profiles as functions of travel. The optimization code “E04UCF”, from the NAG package, is used for this phase of the work.","PeriodicalId":331326,"journal":{"name":"Computational Methods for Solution of Inverse Problems in Mechanics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1998-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129960010","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Web processing requires that the web is stretched under tension between two processing rolls. When the processing rolls are misaligned or moved relative to each other, a web deflection is introduced which causes the web stress to change. Small roll misalignments or movements may cause large web stresses, yielding, and failure. The web stress distribution and deflection can only be found if the buckled and unbuckled portions of the web are determined. This paper presents the assumptions and analytical method needed to find the boundary edge between the buckled and unbuckled portions of the web. Also, the method for analyzing the unbuckled portion of the web to determine the web end deflection, end rotation, and maximum stress is included.
{"title":"Linear Buckle Analysis for Partially Buckled Webs","authors":"Mark V. Loen","doi":"10.1115/imece1998-0225","DOIUrl":"https://doi.org/10.1115/imece1998-0225","url":null,"abstract":"\u0000 Web processing requires that the web is stretched under tension between two processing rolls. When the processing rolls are misaligned or moved relative to each other, a web deflection is introduced which causes the web stress to change. Small roll misalignments or movements may cause large web stresses, yielding, and failure.\u0000 The web stress distribution and deflection can only be found if the buckled and unbuckled portions of the web are determined. This paper presents the assumptions and analytical method needed to find the boundary edge between the buckled and unbuckled portions of the web. Also, the method for analyzing the unbuckled portion of the web to determine the web end deflection, end rotation, and maximum stress is included.","PeriodicalId":331326,"journal":{"name":"Computational Methods for Solution of Inverse Problems in Mechanics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1998-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128214198","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper addresses an inverse vibration problem for cantilever composite plates. A two-step procedure is proposed to estimate the elastic properties of the material. First, simple formulas are used to provide an initial estimate given experimentally determined natural frequencies. A very accurate Rayleigh-Ritz model is then used to refine these estimates and obtain good agreement between predictions and experimental values. Imperfect support conditions at the root affect the dynamics of the plate significantly. Neglecting this effect leads to unrealistic estimates of the material properties. A procedure to identify the stiffness of the supports is presented and demonstrated on several examples.
{"title":"Identification Problems for Vibrating Composite Plates","authors":"S. Abrate, M. Perry","doi":"10.1115/imece1998-0224","DOIUrl":"https://doi.org/10.1115/imece1998-0224","url":null,"abstract":"\u0000 This paper addresses an inverse vibration problem for cantilever composite plates. A two-step procedure is proposed to estimate the elastic properties of the material. First, simple formulas are used to provide an initial estimate given experimentally determined natural frequencies. A very accurate Rayleigh-Ritz model is then used to refine these estimates and obtain good agreement between predictions and experimental values. Imperfect support conditions at the root affect the dynamics of the plate significantly. Neglecting this effect leads to unrealistic estimates of the material properties. A procedure to identify the stiffness of the supports is presented and demonstrated on several examples.","PeriodicalId":331326,"journal":{"name":"Computational Methods for Solution of Inverse Problems in Mechanics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1998-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131633890","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This work presents some recent results on the solution of shallow-water flows using the finite element method and the optimal control method for distributed systems. The objective is to identify the friction coefficient for steady and unsteady flows, such as the numerical solution is close to the measurement data. We investigate the following aspects: the finite element discretization, the regularization of the cost function and the solution algorithms. Several benchmark tests are presented.
{"title":"Identification of the Friction Coefficient for Steady and Unsteady Shallow-Water Flows","authors":"A. Soulaimani","doi":"10.1115/imece1998-0220","DOIUrl":"https://doi.org/10.1115/imece1998-0220","url":null,"abstract":"\u0000 This work presents some recent results on the solution of shallow-water flows using the finite element method and the optimal control method for distributed systems. The objective is to identify the friction coefficient for steady and unsteady flows, such as the numerical solution is close to the measurement data. We investigate the following aspects: the finite element discretization, the regularization of the cost function and the solution algorithms. Several benchmark tests are presented.","PeriodicalId":331326,"journal":{"name":"Computational Methods for Solution of Inverse Problems in Mechanics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1998-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115171139","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Estimation of tractions and displacements on inaccessible boundaries, such as contact areas of solids, can be regarded as an inverse boundary value problem. In this study finite-element based inverse analysis schemes with regularization were applied to the estimation of the distributions of tractions and displacements on contact areas. The finite element equation was rewritten in terms of unknown boundary values on the contact area using over-prescribed boundary values. This equation was solved for the boundary values on the contact area. Like many other inverse problems, this inverse problem was severely ill-conditioned and the estimated distributions were very sensitive to the over-prescribed boundary values used in the estimation. To overcome the ill-posedness of this boundary value inverse problem, the function expansion method and Tikhonov regularization were introduced in the finite element-based inverse analysis scheme. The number of terms in the function expansion and the smoothing parameter in Tikhonov regularization were regarded as regularization parameters in the inverse analysis. To determine the optimum value of these regularization parameters, the estimated error criterion and the AIC were introduced. The usefulness of the finite element-based inversion scheme was examined by numerical simulations. It was found that the distributions of tractions and displacements can be estimated reasonably even from noisy observations by using the finite-element based inverse analysis schemes with regularization. The optimum value of the regularization parameters can be estimated by the estimated error criterion or by the AIC.
{"title":"Estimation of Distributions of Contact Stressess and Displacements Using Regularization Schemes","authors":"S. Kubo, K. Ohji","doi":"10.1115/imece1998-0219","DOIUrl":"https://doi.org/10.1115/imece1998-0219","url":null,"abstract":"\u0000 Estimation of tractions and displacements on inaccessible boundaries, such as contact areas of solids, can be regarded as an inverse boundary value problem. In this study finite-element based inverse analysis schemes with regularization were applied to the estimation of the distributions of tractions and displacements on contact areas. The finite element equation was rewritten in terms of unknown boundary values on the contact area using over-prescribed boundary values. This equation was solved for the boundary values on the contact area. Like many other inverse problems, this inverse problem was severely ill-conditioned and the estimated distributions were very sensitive to the over-prescribed boundary values used in the estimation. To overcome the ill-posedness of this boundary value inverse problem, the function expansion method and Tikhonov regularization were introduced in the finite element-based inverse analysis scheme. The number of terms in the function expansion and the smoothing parameter in Tikhonov regularization were regarded as regularization parameters in the inverse analysis. To determine the optimum value of these regularization parameters, the estimated error criterion and the AIC were introduced. The usefulness of the finite element-based inversion scheme was examined by numerical simulations. It was found that the distributions of tractions and displacements can be estimated reasonably even from noisy observations by using the finite-element based inverse analysis schemes with regularization. The optimum value of the regularization parameters can be estimated by the estimated error criterion or by the AIC.","PeriodicalId":331326,"journal":{"name":"Computational Methods for Solution of Inverse Problems in Mechanics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1998-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133597988","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper presents an inverse problem methodology for the identification of structural joint characteristics. The underlying theory employs a substructural flexibility method that allows statistical uncertainties in joints and interfaces. The method partitions structures into continuum and localized joint/interface substructural regions. The former are modeled by continuum finite elements and built up with the standard direct-stiffness method. Joint and interface regions are constructed from continuum and/or special elements. The novel aspect of the method is that the coupling of substructures is effected in terms of node-collocated Lagrange multipliers, which leads naturally to a boundary-flexibility formulation. Coupling through interaction forces addresses the concern that delicate localized effects are not masked out in the overall structural model. This heightened sensitivity is significant for inverse problems in which localized properties and uncertainties are deduced indirectly, through a hierarchical “peeling off” process. The method is applied to identify joint flexibilities in a model engine mount, which demonstrates key features of the method and its high-fidelity capability.
{"title":"A Flexibility-Based Inverse Algorithm for Identification of Substructural and Joint Properties","authors":"K. Park, C. Felippa","doi":"10.1115/imece1998-0216","DOIUrl":"https://doi.org/10.1115/imece1998-0216","url":null,"abstract":"\u0000 This paper presents an inverse problem methodology for the identification of structural joint characteristics. The underlying theory employs a substructural flexibility method that allows statistical uncertainties in joints and interfaces. The method partitions structures into continuum and localized joint/interface substructural regions. The former are modeled by continuum finite elements and built up with the standard direct-stiffness method. Joint and interface regions are constructed from continuum and/or special elements. The novel aspect of the method is that the coupling of substructures is effected in terms of node-collocated Lagrange multipliers, which leads naturally to a boundary-flexibility formulation. Coupling through interaction forces addresses the concern that delicate localized effects are not masked out in the overall structural model. This heightened sensitivity is significant for inverse problems in which localized properties and uncertainties are deduced indirectly, through a hierarchical “peeling off” process. The method is applied to identify joint flexibilities in a model engine mount, which demonstrates key features of the method and its high-fidelity capability.","PeriodicalId":331326,"journal":{"name":"Computational Methods for Solution of Inverse Problems in Mechanics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1998-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114630052","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The inverse problem of recovering an unknown boundary function given some information about the (interior) solution is considered. It is assumed that the forward problem is governed by a linear partial differential equation (PDE) and that homogeneous boundary conditions exist on the rest of the boundary. A method developed by Tarantola (Tar87) is used which is based upon statistical inference on finite dimensional random vectors. Within this framework, a method for constructing Gaussian probability density functions which model the forward problem is proposed.
{"title":"A Method for Defining Gaussian Probability Densities for Forward Modeling in Finite Dimensions Using the Method of Tarantola","authors":"C. Clutz, A. Maniatty","doi":"10.1115/imece1998-0218","DOIUrl":"https://doi.org/10.1115/imece1998-0218","url":null,"abstract":"\u0000 The inverse problem of recovering an unknown boundary function given some information about the (interior) solution is considered. It is assumed that the forward problem is governed by a linear partial differential equation (PDE) and that homogeneous boundary conditions exist on the rest of the boundary. A method developed by Tarantola (Tar87) is used which is based upon statistical inference on finite dimensional random vectors. Within this framework, a method for constructing Gaussian probability density functions which model the forward problem is proposed.","PeriodicalId":331326,"journal":{"name":"Computational Methods for Solution of Inverse Problems in Mechanics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1998-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125340493","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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