Author: Roger W. Pryor
Publisher: Mercury Learning and Information
ISBN: 1683925882
Category : Technology & Engineering
Languages : en
Pages : 743
Book Description
COMSOL 5 and MATLAB are valuable software modeling tools for engineers and scientists. This updated edition includes five new models and explores a wide range of models in coordinate systems from 0D to 3D, introducing the numerical analysis techniques employed in COMSOL 5.6 and MATLAB software. The text presents electromagnetic, electronic, optical, thermal physics, and biomedical models as examples. It presents the fundamental concepts in the models and the step-by-step instructions needed to build each model. The companion files include all the built models for each step-by-step example presented in the text and the related animations, as specified. The book is designed to introduce modeling to an experienced engineer or can also be used for upper level undergraduate or graduate courses. FEATURES: Focuses on COMSOL 5.x and MATLAB models that demonstrate the use of concepts for later application in engineering, science, medicine, and biophysics for the development of devices and systems Includes companion files with executable copies of each model and related animations Includes detailed discussions of possible modeling errors and results Uses a step-by-step modeling methodology linked to the Fundamental Laws of Physics. The companion files are also available online by emailing the publisher with proof of purchase at [email protected].
Multiphysics Modeling Using COMSOL 5 and MATLAB
Author: Roger W. Pryor
Publisher: Mercury Learning and Information
ISBN: 1683925882
Category : Technology & Engineering
Languages : en
Pages : 743
Book Description
COMSOL 5 and MATLAB are valuable software modeling tools for engineers and scientists. This updated edition includes five new models and explores a wide range of models in coordinate systems from 0D to 3D, introducing the numerical analysis techniques employed in COMSOL 5.6 and MATLAB software. The text presents electromagnetic, electronic, optical, thermal physics, and biomedical models as examples. It presents the fundamental concepts in the models and the step-by-step instructions needed to build each model. The companion files include all the built models for each step-by-step example presented in the text and the related animations, as specified. The book is designed to introduce modeling to an experienced engineer or can also be used for upper level undergraduate or graduate courses. FEATURES: Focuses on COMSOL 5.x and MATLAB models that demonstrate the use of concepts for later application in engineering, science, medicine, and biophysics for the development of devices and systems Includes companion files with executable copies of each model and related animations Includes detailed discussions of possible modeling errors and results Uses a step-by-step modeling methodology linked to the Fundamental Laws of Physics. The companion files are also available online by emailing the publisher with proof of purchase at [email protected].
Publisher: Mercury Learning and Information
ISBN: 1683925882
Category : Technology & Engineering
Languages : en
Pages : 743
Book Description
COMSOL 5 and MATLAB are valuable software modeling tools for engineers and scientists. This updated edition includes five new models and explores a wide range of models in coordinate systems from 0D to 3D, introducing the numerical analysis techniques employed in COMSOL 5.6 and MATLAB software. The text presents electromagnetic, electronic, optical, thermal physics, and biomedical models as examples. It presents the fundamental concepts in the models and the step-by-step instructions needed to build each model. The companion files include all the built models for each step-by-step example presented in the text and the related animations, as specified. The book is designed to introduce modeling to an experienced engineer or can also be used for upper level undergraduate or graduate courses. FEATURES: Focuses on COMSOL 5.x and MATLAB models that demonstrate the use of concepts for later application in engineering, science, medicine, and biophysics for the development of devices and systems Includes companion files with executable copies of each model and related animations Includes detailed discussions of possible modeling errors and results Uses a step-by-step modeling methodology linked to the Fundamental Laws of Physics. The companion files are also available online by emailing the publisher with proof of purchase at [email protected].
Multiphysics Modeling Using COMSOL?
Author: Roger Pryor
Publisher: Jones & Bartlett Learning
ISBN: 0763779997
Category : Computers
Languages : en
Pages : 872
Book Description
Multiphysics Modeling Using COMSOL? rapidly introduces the senior level undergraduate, graduate or professional scientist or engineer to the art and science of computerized modeling for physical systems and devices. It offers a step-by-step modeling methodology through examples that are linked to the Fundamental Laws of Physics through a First Principles Analysis approach. The text explores a breadth of multiphysics models in coordinate systems that range from 1D to 3D and introduces the readers to the numerical analysis modeling techniques employed in the COMSOL? Multiphysics? software. After readers have built and run the examples, they will have a much firmer understanding of the concepts, skills, and benefits acquired from the use of computerized modeling techniques to solve their current technological problems and to explore new areas of application for their particular technological areas of interest.
Publisher: Jones & Bartlett Learning
ISBN: 0763779997
Category : Computers
Languages : en
Pages : 872
Book Description
Multiphysics Modeling Using COMSOL? rapidly introduces the senior level undergraduate, graduate or professional scientist or engineer to the art and science of computerized modeling for physical systems and devices. It offers a step-by-step modeling methodology through examples that are linked to the Fundamental Laws of Physics through a First Principles Analysis approach. The text explores a breadth of multiphysics models in coordinate systems that range from 1D to 3D and introduces the readers to the numerical analysis modeling techniques employed in the COMSOL? Multiphysics? software. After readers have built and run the examples, they will have a much firmer understanding of the concepts, skills, and benefits acquired from the use of computerized modeling techniques to solve their current technological problems and to explore new areas of application for their particular technological areas of interest.
Modelling in Science and Engineering
Author: Marcus Inácio
Publisher:
ISBN: 9781795702348
Category :
Languages : en
Pages : 255
Book Description
This book gives the reader a brief introduction to the COMSOL Multiphysics software tool. Building COMSOL Multiphysics models in 2D or 3D will help students to consolidate their skills by applying basic theory to the real modelling of tasks that in the recent past would require months of programming and dedicated projects to solve a single problem. The examples illustrated in this book include modelling of heat transfer, the migration of a radioactive species in a channel using the Navier-Stokes equations and a chemical heterogenous reactor. These are problems that tend to be rather abstract until such time as a student applies these fundamental equations in practice. Advanced coupling between phenomena in fields such as electromagnetics with others such as heat transfer and computational fluid flow is made easy in COMSOL Multiphysics. A short introduction to the basics, concepts and techniques will allow the reader to progress rapidly and start developing his/her own models. In the second part of this book, some of the models developed in the first part are used to create model applications that can even run on a mobile phone. About the authors: António de Campos Pereira, PhD. in Physics, is an author and consultant. He is a retired researcher from the Dept. of Physics at Stockholm University. Prof. Isabel Paiva, Ph.D. in Chemical Engeneering, is a researcher at C2TN at IST, the School of Engineering of the University of Lisbon. Marcus Inácio has a B.Sc. in Electrotechnical Engineering and is specialising in the field of Medical Physics at KTH, the Royal Institute of Technology in Stockholm, Sweden. Hugo de Campos Pereira is an environmental engineer from Uppsala University and a Ph.D. student specialising in the sorption of highly fluorinated compounds in soils at the Department of Soil and Environment at SLU, the Swedish University of Agricultural Sciences in Uppsala, Sweden.
Publisher:
ISBN: 9781795702348
Category :
Languages : en
Pages : 255
Book Description
This book gives the reader a brief introduction to the COMSOL Multiphysics software tool. Building COMSOL Multiphysics models in 2D or 3D will help students to consolidate their skills by applying basic theory to the real modelling of tasks that in the recent past would require months of programming and dedicated projects to solve a single problem. The examples illustrated in this book include modelling of heat transfer, the migration of a radioactive species in a channel using the Navier-Stokes equations and a chemical heterogenous reactor. These are problems that tend to be rather abstract until such time as a student applies these fundamental equations in practice. Advanced coupling between phenomena in fields such as electromagnetics with others such as heat transfer and computational fluid flow is made easy in COMSOL Multiphysics. A short introduction to the basics, concepts and techniques will allow the reader to progress rapidly and start developing his/her own models. In the second part of this book, some of the models developed in the first part are used to create model applications that can even run on a mobile phone. About the authors: António de Campos Pereira, PhD. in Physics, is an author and consultant. He is a retired researcher from the Dept. of Physics at Stockholm University. Prof. Isabel Paiva, Ph.D. in Chemical Engeneering, is a researcher at C2TN at IST, the School of Engineering of the University of Lisbon. Marcus Inácio has a B.Sc. in Electrotechnical Engineering and is specialising in the field of Medical Physics at KTH, the Royal Institute of Technology in Stockholm, Sweden. Hugo de Campos Pereira is an environmental engineer from Uppsala University and a Ph.D. student specialising in the sorption of highly fluorinated compounds in soils at the Department of Soil and Environment at SLU, the Swedish University of Agricultural Sciences in Uppsala, Sweden.
COMSOL5 for Engineers
Author: Mehrzad Tabatabaian
Publisher: Mercury Learning and Information
ISBN: 1942270453
Category : Science
Languages : en
Pages : 437
Book Description
COMSOL5 Multiphysics® is one of the most valuable software modeling tools for engineers and scientists. This book, an updated edition of the previously published, COMSOL for Engineers, covers COMSOL5 which now includes a revolutionary tool, the Application Builder. This component enables users to build apps based on COMSOL models that can be run on almost any operating system (Windows, MAC, mobile/iOS, etc.). Designed for engineers from various disciplines, the book introduces multiphysics modeling techniques and examples accompanied by practical applications using COMSOL5.x. The main objective is to introduce readers to use COMSOL as an engineering tool for modeling, by solving examples that could become a guide for modeling similar or more complicated problems. The book provides a collection of examples and modeling guidelines through which readers can build their own models. The mathematical fundamentals, engineering principles, and design criteria are presented as integral parts of the examples. At the end of chapters are references that contain more in-depth physics, technical information, and data; these are referred to throughout the book and used in the examples. COMSOL5 for Engineers could be used to complement another text that provides background training in engineering computations and methods. Exercises are provided at the end of the text for use in adoption situations. Features: •Expands the Finite Element Method (FEM) theory and adds more examples from the original edition •Outlines the new features in COMSOL5, the graphical user interface (GUI), and how to build a COMSOL app for models •Includes apps for selected model examples-with parameterization of these models •Features new and modified, solved model examples, in addition to the models provided in the original edition •Companion disc with executable copies of each model and their related animations eBook Customers: Companion files are available for downloading with order number/proof of purchase by writing to the publisher at [email protected].
Publisher: Mercury Learning and Information
ISBN: 1942270453
Category : Science
Languages : en
Pages : 437
Book Description
COMSOL5 Multiphysics® is one of the most valuable software modeling tools for engineers and scientists. This book, an updated edition of the previously published, COMSOL for Engineers, covers COMSOL5 which now includes a revolutionary tool, the Application Builder. This component enables users to build apps based on COMSOL models that can be run on almost any operating system (Windows, MAC, mobile/iOS, etc.). Designed for engineers from various disciplines, the book introduces multiphysics modeling techniques and examples accompanied by practical applications using COMSOL5.x. The main objective is to introduce readers to use COMSOL as an engineering tool for modeling, by solving examples that could become a guide for modeling similar or more complicated problems. The book provides a collection of examples and modeling guidelines through which readers can build their own models. The mathematical fundamentals, engineering principles, and design criteria are presented as integral parts of the examples. At the end of chapters are references that contain more in-depth physics, technical information, and data; these are referred to throughout the book and used in the examples. COMSOL5 for Engineers could be used to complement another text that provides background training in engineering computations and methods. Exercises are provided at the end of the text for use in adoption situations. Features: •Expands the Finite Element Method (FEM) theory and adds more examples from the original edition •Outlines the new features in COMSOL5, the graphical user interface (GUI), and how to build a COMSOL app for models •Includes apps for selected model examples-with parameterization of these models •Features new and modified, solved model examples, in addition to the models provided in the original edition •Companion disc with executable copies of each model and their related animations eBook Customers: Companion files are available for downloading with order number/proof of purchase by writing to the publisher at [email protected].
Introduction to Chemical Engineering Computing
Author: Bruce A. Finlayson
Publisher: John Wiley & Sons
ISBN: 1118309588
Category : Technology & Engineering
Languages : en
Pages : 415
Book Description
Step-by-step instructions enable chemical engineers to master key software programs and solve complex problems Today, both students and professionals in chemical engineering must solve increasingly complex problems dealing with refineries, fuel cells, microreactors, and pharmaceutical plants, to name a few. With this book as their guide, readers learn to solve these problems using their computers and Excel®, MATLAB, Aspen Plus, and COMSOL Multiphysics. Moreover, they learn how to check their solutions and validate their results to make sure they have solved the problems correctly. Now in its Second Edition, Introduction to Chemical Engineering Computing is based on the author's firsthand teaching experience. As a result, the emphasis is on problem solving. Simple introductions help readers become conversant with each program and then tackle a broad range of problems in chemical engineering, including: Equations of state Chemical reaction equilibria Mass balances with recycle streams Thermodynamics and simulation of mass transfer equipment Process simulation Fluid flow in two and three dimensions All the chapters contain clear instructions, figures, and examples to guide readers through all the programs and types of chemical engineering problems. Problems at the end of each chapter, ranging from simple to difficult, allow readers to gradually build their skills, whether they solve the problems themselves or in teams. In addition, the book's accompanying website lists the core principles learned from each problem, both from a chemical engineering and a computational perspective. Covering a broad range of disciplines and problems within chemical engineering, Introduction to Chemical Engineering Computing is recommended for both undergraduate and graduate students as well as practicing engineers who want to know how to choose the right computer software program and tackle almost any chemical engineering problem.
Publisher: John Wiley & Sons
ISBN: 1118309588
Category : Technology & Engineering
Languages : en
Pages : 415
Book Description
Step-by-step instructions enable chemical engineers to master key software programs and solve complex problems Today, both students and professionals in chemical engineering must solve increasingly complex problems dealing with refineries, fuel cells, microreactors, and pharmaceutical plants, to name a few. With this book as their guide, readers learn to solve these problems using their computers and Excel®, MATLAB, Aspen Plus, and COMSOL Multiphysics. Moreover, they learn how to check their solutions and validate their results to make sure they have solved the problems correctly. Now in its Second Edition, Introduction to Chemical Engineering Computing is based on the author's firsthand teaching experience. As a result, the emphasis is on problem solving. Simple introductions help readers become conversant with each program and then tackle a broad range of problems in chemical engineering, including: Equations of state Chemical reaction equilibria Mass balances with recycle streams Thermodynamics and simulation of mass transfer equipment Process simulation Fluid flow in two and three dimensions All the chapters contain clear instructions, figures, and examples to guide readers through all the programs and types of chemical engineering problems. Problems at the end of each chapter, ranging from simple to difficult, allow readers to gradually build their skills, whether they solve the problems themselves or in teams. In addition, the book's accompanying website lists the core principles learned from each problem, both from a chemical engineering and a computational perspective. Covering a broad range of disciplines and problems within chemical engineering, Introduction to Chemical Engineering Computing is recommended for both undergraduate and graduate students as well as practicing engineers who want to know how to choose the right computer software program and tackle almost any chemical engineering problem.
Multiphysics Modeling Using COMSOL®5 and MATLAB®
Author: Roger W. Pryor
Publisher:
ISBN: 9781523100156
Category : COMSOL Multiphysics
Languages : en
Pages : 594
Book Description
"COMSOL 5 and MATLAB are valuable software modeling tools for engineers and scientists. This book explores a wide range of models in coordinate systems from 0D to 3D and introduces the numerical analysis techniques employed in COMSOL 5 (including Application Builder) and MATLAB software. The text presents electromagnetic, electronic, optical, thermal physics, and biomedical models as examples. It presents the fundamental concepts in the models and the step-by-step instructions needed to build each model. The book is designed to introduce modeling to an experienced engineer or can also be used for upper level undergraduate or graduate courses." --
Publisher:
ISBN: 9781523100156
Category : COMSOL Multiphysics
Languages : en
Pages : 594
Book Description
"COMSOL 5 and MATLAB are valuable software modeling tools for engineers and scientists. This book explores a wide range of models in coordinate systems from 0D to 3D and introduces the numerical analysis techniques employed in COMSOL 5 (including Application Builder) and MATLAB software. The text presents electromagnetic, electronic, optical, thermal physics, and biomedical models as examples. It presents the fundamental concepts in the models and the step-by-step instructions needed to build each model. The book is designed to introduce modeling to an experienced engineer or can also be used for upper level undergraduate or graduate courses." --
Comsol Heat Transfer Models
Author: Layla S. Mayboudi
Publisher: Multiphysics Modeling
ISBN: 9781683922117
Category : Computers
Languages : en
Pages : 400
Book Description
This book guides the reader through the process of model creation for heat transfer analysis with the finite element method. The book describes thermal imaging experiments that demonstrate how such models can be validated. It presents application examples, such as heating water in a kettle, to basement insulation, a heated seat, molten rock, pipe flow, and an innovative extended surface. A companion disc provides the files so models can be run (using COMSOL or other software) in order to observe real-world behavior of the applications. Historical background information is provided to show the progression of heat transfer science and mathematical modeling from the earliest developments to the most recent advances in technology. Features: Includes example models that enable the reader to implement conceptual material in practical scenarios with broad industrial applications Includes companion files with models and geometry files created with COMSOL Multiphysics(R) or imported from a third-party CAD tool.
Publisher: Multiphysics Modeling
ISBN: 9781683922117
Category : Computers
Languages : en
Pages : 400
Book Description
This book guides the reader through the process of model creation for heat transfer analysis with the finite element method. The book describes thermal imaging experiments that demonstrate how such models can be validated. It presents application examples, such as heating water in a kettle, to basement insulation, a heated seat, molten rock, pipe flow, and an innovative extended surface. A companion disc provides the files so models can be run (using COMSOL or other software) in order to observe real-world behavior of the applications. Historical background information is provided to show the progression of heat transfer science and mathematical modeling from the earliest developments to the most recent advances in technology. Features: Includes example models that enable the reader to implement conceptual material in practical scenarios with broad industrial applications Includes companion files with models and geometry files created with COMSOL Multiphysics(R) or imported from a third-party CAD tool.
Multiphysics Modeling With Finite Element Methods
Author: William B J Zimmerman
Publisher: World Scientific Publishing Company
ISBN: 9813106735
Category : Technology & Engineering
Languages : en
Pages : 434
Book Description
Finite element methods for approximating partial differential equations that arise in science and engineering analysis find widespread application. Numerical analysis tools make the solutions of coupled physics, mechanics, chemistry, and even biology accessible to the novice modeler. Nevertheless, modelers must be aware of the limitations and difficulties in developing numerical models that faithfully represent the system they are modeling.This textbook introduces the intellectual framework for modeling with Comsol Multiphysics, a package which has unique features in representing multiply linked domains with complex geometry, highly coupled and nonlinear equation systems, and arbitrarily complicated boundary, auxiliary, and initial conditions. But with this modeling power comes great opportunities and great perils.Progressively, in the first part of the book the novice modeler develops an understanding of how to build up complicated models piecemeal and test them modularly. The second part of the book introduces advanced analysis techniques. The final part of the book deals with case studies in a broad range of application areas including nonlinear pattern formation, thin film dynamics and heterogeneous catalysis, composite and effective media for heat, mass, conductivity, and dispersion, population balances, tomography, multiphase flow, electrokinetic, microfluidic networks, plasma dynamics, and corrosion chemistry.As a revision of Process Modeling and Simulation with Finite Element Methods, this book uses the very latest features of Comsol Multiphysics. There are new case studies on multiphase flow with phase change, plasma dynamics, electromagnetohydrodynamics, microfluidic mixing, and corrosion. In addition, major improvements to the level set method for multiphase flow to ensure phase conservation is introduced.
Publisher: World Scientific Publishing Company
ISBN: 9813106735
Category : Technology & Engineering
Languages : en
Pages : 434
Book Description
Finite element methods for approximating partial differential equations that arise in science and engineering analysis find widespread application. Numerical analysis tools make the solutions of coupled physics, mechanics, chemistry, and even biology accessible to the novice modeler. Nevertheless, modelers must be aware of the limitations and difficulties in developing numerical models that faithfully represent the system they are modeling.This textbook introduces the intellectual framework for modeling with Comsol Multiphysics, a package which has unique features in representing multiply linked domains with complex geometry, highly coupled and nonlinear equation systems, and arbitrarily complicated boundary, auxiliary, and initial conditions. But with this modeling power comes great opportunities and great perils.Progressively, in the first part of the book the novice modeler develops an understanding of how to build up complicated models piecemeal and test them modularly. The second part of the book introduces advanced analysis techniques. The final part of the book deals with case studies in a broad range of application areas including nonlinear pattern formation, thin film dynamics and heterogeneous catalysis, composite and effective media for heat, mass, conductivity, and dispersion, population balances, tomography, multiphase flow, electrokinetic, microfluidic networks, plasma dynamics, and corrosion chemistry.As a revision of Process Modeling and Simulation with Finite Element Methods, this book uses the very latest features of Comsol Multiphysics. There are new case studies on multiphase flow with phase change, plasma dynamics, electromagnetohydrodynamics, microfluidic mixing, and corrosion. In addition, major improvements to the level set method for multiphase flow to ensure phase conservation is introduced.
Modeling and Simulation of Chemical Process Systems
Author: Nayef Ghasem
Publisher: CRC Press
ISBN: 1351339338
Category : Science
Languages : en
Pages : 472
Book Description
In this textbook, the author teaches readers how to model and simulate a unit process operation through developing mathematical model equations, solving model equations manually, and comparing results with those simulated through software. It covers both lumped parameter systems and distributed parameter systems, as well as using MATLAB and Simulink to solve the system model equations for both. Simplified partial differential equations are solved using COMSOL, an effective tool to solve PDE, using the fine element method. This book includes end of chapter problems and worked examples, and summarizes reader goals at the beginning of each chapter.
Publisher: CRC Press
ISBN: 1351339338
Category : Science
Languages : en
Pages : 472
Book Description
In this textbook, the author teaches readers how to model and simulate a unit process operation through developing mathematical model equations, solving model equations manually, and comparing results with those simulated through software. It covers both lumped parameter systems and distributed parameter systems, as well as using MATLAB and Simulink to solve the system model equations for both. Simplified partial differential equations are solved using COMSOL, an effective tool to solve PDE, using the fine element method. This book includes end of chapter problems and worked examples, and summarizes reader goals at the beginning of each chapter.
Introduction to Software for Chemical Engineers, Second Edition
Author: Mariano Martín Martín
Publisher: CRC Press
ISBN: 0429835426
Category : Computers
Languages : en
Pages : 959
Book Description
The field of Chemical Engineering and its link to computer science is in constant evolution and new engineers have a variety of tools at their disposal to tackle their everyday problems. Introduction to Software for Chemical Engineers, Second Edition provides a quick guide to the use of various computer packages for chemical engineering applications. It covers a range of software applications from Excel and general mathematical packages such as MATLAB and MathCAD to process simulators, CHEMCAD and ASPEN, equation-based modeling languages, gProms, optimization software such as GAMS and AIMS, and specialized software like CFD or DEM codes. The different packages are introduced and applied to solve typical problems in fluid mechanics, heat and mass transfer, mass and energy balances, unit operations, reactor engineering, process and equipment design and control. This new edition offers a wider view of packages including open source software such as R, Python and Julia. It also includes complete examples in ASPEN Plus, adds ANSYS Fluent to CFD codes, Lingo to the optimization packages, and discusses Engineering Equation Solver. It offers a global idea of the capabilities of the software used in the chemical engineering field and provides examples for solving real-world problems. Written by leading experts, this book is a must-have reference for chemical engineers looking to grow in their careers through the use of new and improving computer software. Its user-friendly approach to simulation and optimization as well as its example-based presentation of the software, makes it a perfect teaching tool for both undergraduate and master levels.
Publisher: CRC Press
ISBN: 0429835426
Category : Computers
Languages : en
Pages : 959
Book Description
The field of Chemical Engineering and its link to computer science is in constant evolution and new engineers have a variety of tools at their disposal to tackle their everyday problems. Introduction to Software for Chemical Engineers, Second Edition provides a quick guide to the use of various computer packages for chemical engineering applications. It covers a range of software applications from Excel and general mathematical packages such as MATLAB and MathCAD to process simulators, CHEMCAD and ASPEN, equation-based modeling languages, gProms, optimization software such as GAMS and AIMS, and specialized software like CFD or DEM codes. The different packages are introduced and applied to solve typical problems in fluid mechanics, heat and mass transfer, mass and energy balances, unit operations, reactor engineering, process and equipment design and control. This new edition offers a wider view of packages including open source software such as R, Python and Julia. It also includes complete examples in ASPEN Plus, adds ANSYS Fluent to CFD codes, Lingo to the optimization packages, and discusses Engineering Equation Solver. It offers a global idea of the capabilities of the software used in the chemical engineering field and provides examples for solving real-world problems. Written by leading experts, this book is a must-have reference for chemical engineers looking to grow in their careers through the use of new and improving computer software. Its user-friendly approach to simulation and optimization as well as its example-based presentation of the software, makes it a perfect teaching tool for both undergraduate and master levels.