COMSOL BV provides software solutions for multiphysics modeling. It is a fast growing high tech engineering software company with a proven track record and a vision as a future leader of the industry. The company was founded in July 1986 in Stockholm, Sweden and throughout the years has grown to include offices in Brazil, China, Denmark, Finland, France, Germany, India, Italy, the Netherlands, Norway, Russia, Switzerland, the United Kingdom, and the U.S.A.
COMSOL Products
In 1998 COMSOL published the first version of its flagship product, COMSOL Multiphysics® – an engineering, design, and finite element analysis software environment for the modeling and simulation of any physics-based system. In subsequent years, the product line was expanded to include a suite of discipline-specific add-on modules for Structural Mechanics, High and Low Frequency Electromagnetics, Fluid Flow, Heat Transfer, Chemical Reactions, MEMS, Acoustics, and more. Applications for the software are found in virtually every area of science and engineering. Interfacing tools enable the integration of COMSOL Multiphysics® simulations with all major technical computing and CAD tools on the CAE market.
COMSOL Multiphysics® contains functionality for building simulation applications, the Application Builder. This allows simulation experts to create intuitive and very specific user interfaces for their otherwise general computer models — ready-to-use custom applications. The applications can be deployed with the addition of the COMSOL Server™ or COMSOL Compiler™ products. Both products allow experts to share finished applications with their design teams, manufacturing departments, process operators, test laboratories, customers, and clients worldwide. COMSOL Server™ is an application management tool where applications safely and securely can be deployed, either through an internal network or across the web. COMSOL Compiler™ is an add-on to COMSOL Multiphysics® that enables creating standalone applications, which can be run on a desktop or laptop computer without COMSOL Multiphysics® or COMSOL Server™.
COMSOL’s customers
COMSOL products help researchers and engineers design and develop better products. Leading technical enterprises, research labs, and universities use this simulation tools to increase productivity and problem solving capacity. COMSOL’s customers make our cars and aircrafts safer and more energy efficient, they enhance the reception to our cell phones, they search for new energy sources, explore the universe, develop medical equipment enabling more accurate diagnosis, and they educate the next generation of scientists.
COMSOL events
COMSOL offers free online presentations featuring live software demonstrations, a range of topics, and guest industry speakers. Participate in the Q&A during the live event or access archived webinars later. To help users expand their knowledge of COMSOL Multiphysics with advanced modeling techniques, the company also offers general intensive courses as well as trainings devoted to a specific science/engineering area.
De nieuwste versie van de multifysische simulatiesoftware introduceert een Electric Discharge Module, GPU-versnelde simulaties en updates voor een hogere modelleerproductiviteit. Lees verder
When a ray of collimated light, such as from a laser, is incident upon a semitransparent medium, it can experience both absorption and scattering. This means that the incident light is both converted to thermal energy and redirected. Under certain assumptions, these phenomena can be modeled using a diffusive approximation in the COMSOL Multiphysics® software. This modeling approach has applications in laser heating of living tissue as well as materials processing. Let’s learn more!
In microfluidic systems, fluid flow is always laminar. This is both a benefit and a burden — a benefit because the flow field is stationary, and a burden because species mixing occurs primarily by diffusion, which can be time-consuming. A simple way to mix chemical species in a microfluidic chip is to use a serpentine channel structure. Using the COMSOL Multiphysics® software, the periodicity of such a structure can be exploited to determine the required channel length that will ensure that the chemical species leaves the system well mixed. Read more
Eigenfrequency analysis is an integral part of the numerical modeling toolkit. The eigenmodes of a linear system often have distinct qualitative characteristics and evolve differently over a parameter range, such as frequency. We are often asked if there is a way to keep track of and categorize these families of eigenmode solutions over the parameter sweep. In this blog post, we will demonstrate how to do so using the mode overlap integral in the COMSOL Multiphysics® software. Read more
NASA combines thermal modeling and experimental testing to find the best compressor design for the system that keeps the air breathable at the International Space Station. Read more
The katana is a legendary sword used by the samurai several hundred centuries ago. It is perhaps most recognizable for its curved shape and its remarkably sharp single edge. In this blog post, we will go over how to build a simple model of a katana using the COMSOL Multiphysics® software and simulate a differential hardening process to explore some of its features. Read more
To demonstrate how the COMSOL Multiphysics® software can be used for modeling semiconductor manufacturing equipment, processes, and devices, we create example models and other guides on a regular basis. Browse example models and see suggested add-on modules in this resource collection. Read more
Structural contact modeling is a highly nonlinear problem. As surfaces come in and out of contact, load paths and stress states will abruptly change. The computational solvers in the COMSOL Multiphysics® software are designed to work with sufficiently smooth solutions, so solving such models is inherently challenging. To efficiently achieve a converged solution, most contact models will require some changes to the default model settings. Read more [Picture source: COMSOL, Inc. on youtube.com]
Composites are widely used in industrial applications. Compared to the traditional monolithic materials, composites can have specialized material properties due to the customization of constituents, making them versatile and applicable to many different industries, such as in areas like aerospace engineering and biomedical engineering. Homogenization techniques are needed to numerically compute the material properties of composites and can be used for the customization and design of versatile materials. In this blog post, we are going to look at a simulation app developed in the COMSOL Multiphysics® software using the Application Builder that can be used for composite material design and material homogenization. Read more
When you speak up in a virtual meeting, recite voice commands into smart devices, or talk over the phone, there’s a good chance it’s MEMS technology that picks up your voice. This is due to the frequent use of this solid-state semiconductor technology in creating small speakers that produce high-quality sound. In this blog post, we explore the benefits that MEMS bring to microphones, the challenges that come with producing MEMS microphones, and how modeling and simulation can help make the design process of these microphones more efficient. We also discuss the latest advancements in modern microspeakers driven by MEMS technology. Read more
I watched the latest Top Gun movie, Top Gun: Maverick a few weeks ago. It is a really awesome movie, and it is also cutting edge from an engineering standpoint. The movie starts with Maverick, played by Tom Cruise, working as a test pilot preparing to fly a new secretly developed airplane, the Darkstar, capable of flying Mach 10. Simultaneously, Rear Admiral Chester Cain, played by Ed Harris, makes his way to the test base to shut down the program — considering the Darkstar had not yet been able to fly Mach 10, which was a requirement for the project to continue — but Maverick takes action before Cain can do so. Just as Cain arrives to the base, Maverick takes off with the Darkstar to make a try for Mach 10. A great start! Read more
HVAC systems do more than provide the smooth, chilled air that flows when the temperature outside rises. Within these systems, air moves through filters to ensure high air quality. With clean air at stake, modeling and simulation can be used to gain an in-depth understanding of the physics behind the behavior of air as it moves through a filter… Read more
Mechanical systems often contain components that exhibit nonlinear material behavior. Examples include large elastic deformations in seals and gaskets, strain-rate dependence and hysteresis during cyclic loading in rubbers and soft biological tissues, and elastoplastic flow and creep in metals. Together with its Nonlinear Structural Materials Module add-on, the COMSOL Multiphysics® software contains more than one hundred built-in material models that can be used for modeling highly complex material behavior. However, a drawback with these — often phenomenological — models is that they can contain a large number of material parameters, which need to be calibrated for each specific material in order to obtain accurate modeling predictions. In today’s blog post, we will demonstrate how these parameters can be estimated from experimental data obtained from common material tests using nonlinear least-squares minimization techniques. Read more
To demonstrate how the COMSOL Multiphysics® software can be used for modeling hydrogen and energy production through electrolyzers, fuel cells, and related technology, we create example models and other guides on a regular basis. Browse example models and see suggested add-on modules in this resource collection. Read more
Composite materials are heterogeneous materials composed of at least two constituent materials. Among the different types of composite materials, layered composite materials are quite common and are widely used for aircraft, spacecraft, wind turbine, automobile, marine, buildings, and safety equipment use cases. The Composite Materials Module, an add-on to the COMSOL Multiphysics® software, includes built-in features and functionality specifically designed for studying layered composite structures. Fiber-reinforced polymers, particulate-reinforced polymers, laminated plates, and sandwich panels are a few common examples of layered composite materials. Read more
Can we “see” sound? Not directly, but we can come close. By changing our perspective, we can learn a lot about the nature of acoustics. One way to observe acoustics phenomena is by studying standing waves in a solid medium known as a Chladni plate. A special technique creates patterns on the plate that reveal sound’s physical nature. Read more and watch the video
The COMSOL Conference 2023 returned in person this year in Munich, Germany, where the multiphysics simulation community formed new connections, learned about the latest advancements in the COMSOL Multiphysics® software, and drew inspiration from each other’s innovative work. During the three-day event, engineers, researchers, and scientists across industries showcased their findings through poster and slideshow presentations. Six teams earned special recognition for their work, receiving Best Paper and Best Poster awards. Read more
Why are superconductors and their applications such a challenging subject to teach? Francesco Grilli, a researcher and professor from the Karlsruhe Institute of Technology (KIT), has an idea as to why — and a solution to the problem. By building simulation applications and making them available in a web browser, Grilli helps present this complex topic in a captivating way to encourage students to stay focused and interested in learning more about superconductivity. (You, fellow COMSOL Blog reader, can also access the apps via the link toward the end of the post!) Read more
Does a new design come from a person or a process? We may imagine that innovators envision a design and then pick up their pencils to bring their idea to life. Of course, designers and engineers today might not use pencils, and they might not have a final design idea in mind. Instead, some designers use a process that comes up with new ideas for them — a methodology known as generative design. Read more
Dimples are famously crucial for the aerodynamic properties of a golf ball: They generate a turbulent flow that reduces the ball’s drag. However, doesn’t this sound counterintuitive? In general, smooth objects are more aerodynamic than rough ones. In today’s blog post, we will dig into the details of this apparent paradox; learn how to use this knowledge to model the trajectory of a golf ball with the COMSOL Multiphysics® software; and, finally, find the best angle to hit the ball. Keep reading for a hole in one… Read more
Multiphysics modeling and simulation accelerates scientific research and increases the understanding of underlying phenomena, making it suited for use in engineering education. In this panel discussion, speakers Dr. Harold Ruiz from the University of Leicester and Dr. James Bowen from the Open University will share how they are using the COMSOL Multiphysics® software as an academic teaching aid. Read more
Semiconductor manufacturing, particle physics research, and other valuable processes occur in high-vacuum or ultra-high-vacuum (HV/UHV) conditions. To help develop a better ionization gauge for measuring pressure in HV/UHV environments, instrument manufacturer INFICON of Liechtenstein used multiphysics modeling to test and refine their impressive new design. Read more
Check out the new COMSOL Learning Center! You’ll find multipart courses, articles, videos, modeling exercises, model files, and step-by-step instructions — all designed to help you get the most out of the COMSOL® software! Read more
The COMSOL Multiphysics® software is well suited for modeling RF heating, where one needs to solve for both the electromagnetic fields and the temperature distribution over time. Although you might think that you need the RF Module for all types of problems involving RF heating, you can often use either the RF Module or the AC/DC Module, and you can use several different physics interfaces for some typical problems in this area. Read more
Have you ever wondered how to create a simulation mesh out of data obtained by 3D imaging techniques? In this blog post, we will explain how to do so using the COMSOL Multiphysics® software. This topic expands on the theme of modeling irregular shapes, which we have explored in past blog posts. The processes we will discuss here are, in part, already used when setting up verification studies for topology optimization results. Nevertheless, in this blog post we will generalize the workflow to suit data obtained by 3D imaging techniques. Read more
Topology optimization is associated with extreme design freedom and thus extreme performance, but the resulting designs are often incompatible with conventional manufacturing techniques. The development of manufacturing constraints for topology optimization is an active research topic. The COMSOL Multiphysics® software supports milling constraints, and in this blog post we will explain how to use such functionality and show examples. Read more
For many structural engineers, beam theory is a popular analysis tool. Using the equations can be beneficial when considering structural behavior, as they are easy to apply and provide useful results. However, it’s also known that due to the simplicity and convenience, beam theory is applied even when some underlying assumption may not hold up too well. This blog post investigates one such case, where there are severe pitfalls of using beam theory, and the true structural behavior is surprisingly different. Read more
Fuel cells are one of the most talked about new technologies in the clean energy domain. Fuel cells generate electricity via electrochemical reactions involving hydrogen and oxygen, where the sum reaction yields hydrogen oxidation and oxygen reduction. Simply put, if a fuel cell has a steady supply of hydrogen and oxygen, it will generate electricity. Furthermore, the byproduct generated in this process is water, thereby making it a “clean fuel”, which doesn’t generate carbon dioxide or toxic byproducts. Read more
It’s quite common to use focused laser light to rapidly heat materials for various purposes, including within the semiconductor processing industry. Here, we will look at a Gaussian profile laser beam with periodically pulsed intensity, heating up two different semitransparent materials deposited onto a silicon substrate. To model this, we will solve a multiphysics modeling problem using the temperature field and the Beer–Lambert law. Let’s further explore the model and see how to set it up… Read more
Calcium carbonates produced by sea creatures help to maintain the ocean’s alkalinity — and also serve as a natural sink for anthropogenic carbon dioxide. To better understand essential but obscure deep-sea galvanization processes, Olivier Sulpis of Utrecht University developed an innovative 3D reactive-transport model that shows how seashells help preserve calcite grains in seafloor sediments. Read more
Shape optimization can be used to improve designs in many different physics areas. In this blog post, we will focus on shape optimization in wave optics. We will go over the shape optimization features in the COMSOL Multiphysics® software and show what you can expect to achieve when these features are used for wave optics applications. Read more
COMSOL Multiphysics® version 6.1 introduces new functionality for detached eddy simulation, thermal analysis of satellites, winding layouts for electric motors, and robust mechanical contact. A new interface enables the analysis of battery packs with several hundred cells. Simulation of acoustically driven flows is made possible by a new acoustic streaming interface. New mesh repair tools for the handling of misaligned models provide an alternative to traditional CAD repair and defeaturing. Direct modeling operations make it possible to perform parametric sweeps of imported CAD models and optimize them. New functionality for including direct shadows in visualizations provides enhanced depth perception. Read more and watch the video
A multidisciplinary team worked together to design and install an aboveground rainwater drainage system in Copenhagen, Denmark. The resulting structure provides an ingenious system for dispersing rainwater runoff from nearby rooftops and mitigating noise pollution. Watch the video
Tectonic Audio Labs created a state-of-the-art balanced mode radiator speaker using electromagnetics, mechanical, and acoustics simulation. The speaker was implemented into a virtual reality (VR) headset for Valve Corporation and is now regarded as the gold standard for VR audio. Read more
Rather than coining the phrase “x-ray vision,” perhaps comic book writers should have called it “terahertz vision” instead. The precise blend of electromagnetic and material properties in the terahertz spectrum enables efficient applications such as simultaneous noncontact package inspection with spectroscopic characterization — to determine, for example, whether a package contains baking flour or an illegal or otherwise dangerous substance. This can be determined without opening the package and while returning images of the package contents with submillimeter precision. Readmore [Source: photonics.com]
Creating clean energy alternatives for energy sources that generate greenhouse gas is crucial if we are to avoid excessive global heating and the rising climate crisis. A viable, renewable energy source are ocean tides, which, unlike solar and wind resources, are not weather dependent, making them predictable. Tidal energy can be harnessed by deploying tidal turbines into tidal streams. This process is the foundation of the MeyGen project, a massive renewable energy project that is planned to be the world’s largest tidal energy plant. Recently, an array of tidal turbines has been deployed at this project site, where tides flowing between the Atlantic Ocean and North Sea are funneled into a high-energy stream between mainland Scotland and the Island of Stroma. Read more
Simulations One of the common objectives of electromagnetics simulation is to extract lumped quantities that can be used in electric circuit simulations. Keep reading to get step-by-step instructions on how to perform such a task using the COMSOL Multiphysics® software… Read more
Graphene and other 2D materials are the focus of intense research and application interests due to their promising properties. In this blog post, we are going to use the example of a graphene-based THz metamaterial perfect absorber to demonstrate how to model 2D materials accurately and efficiently in high-frequency electromagnetics. The discussed techniques are equally applicable to modeling other thin layers, such as coatings on optical devices. Read more
Lithium-ion (Li-ion) batteries are used to power a variety of devices, from toys and drones to cellphones and laptops to medical equipment and electric cars. To efficiently power such devices, the temperature distribution inside operating Li-ion batteries needs to be controlled, as deviation from optimal operating temperatures can cause a loss, or failure, in their performance. One way to analyze the temperature distribution in Li-ion batteries is with multiphysics simulation. Read more
Two professional chefs stand in a classroom, closely observing a soft-boiled egg. What may initially sound like a cooking class is actually part of a physics course offered at the Technische Universiteit Eindhoven (TU/e) in the Netherlands. Read more
The power of multiphysics simulation reaches all elements of product and process design projects with the introduction of the Model Manager server, which facilitates the collaboration between stakeholders and colleagues working on a modeling and simulation project. Readmore
Microlithography lenses are used to project the image of an integrated circuit onto a silicon substrate. This tutorial demonstrates how to create a 21-element fused silica lens which has a NA of 0.56 which is designed to be used at a wavelength of 248nm. The lens, which has a total length of 1 meter, has a magnification of -0.25 with excellent image quality over a 23.4mm image circle. Download the application files
Polar Night Energy, a startup in Finland, has developed technology for warming up buildings with solar-generated heat stored in sand. The team uses thermal modeling to optimize the design of their heat storage and distribution systems, which are helping Finnish cities reduce their consumption of nonrenewable heating fuels. Read more
There has recently been a lot of controversy online regarding an entertaining video that tries to address a big misconception about electricity: that electrons carry the energy in an electrical circuit. Although the video, and many of the response videos, are certainly engaging, there is so much more to the story, and using the COMSOL Multiphysics® software is an excellent way to investigate this theory. Read more
Optical fiber networks, which make up the backbone of the internet, rely on many electrical signal processing devices. Nanoscale silicon photonic network components, such as phase shifters, could boost optical network speed, capacity, and reliability. To design these small but powerful devices, a team at the Swiss Federal Institute of Technology Lausanne (EPFL) uses simulation to optimize both optical and electromechanical performance. Read more
Today, we released a new version of the COMSOL Multiphysics® software, version 6.0! With the release, we bring you the Model Manager, a new core functionality that enables efficient simulation data management and collaboration. Also introduced with the new version is the Uncertainty Quantification Module, a new add-on product for global sensitivity and probabilistic analyses. Additionally, many significant updates have been made to the platform product and add-on products. Get an overview of what’s new in the latest version here. Read more
Do you drool at the very mention of chocolate? If so, you’re a “chocoholic” like me, and Nestlé’s Kit Kat® bar is one of my favorites. For 80 years, people around the globe have devoured this four-piece delight. To ensure every bar of chocolate produced has the same consistency, texture, and taste, the engineers at Nestlé’s Product Technology Centre in York, UK (PTC York) are using simulation to optimize the Kit Kat® bar manufacturing process. Read more
High-performance fusion reactors call for high-performance materials. To optimize a production process for a tungsten material used in a fusion reactor divertor, researchers at Forschungszentrum Jülich GmbH (FZJ), Institute for Energy and Climate Research, and Max Planck Institute for Plasma Physics in Germany turned to multiphysics modeling. Read more
Today, even though most of us aren’t celebrities, we appear on camera more than ever before. Smartphones, computers, and other devices equipped with compact camera modules (CCMs) are almost inescapable — whether or not we feel ready for our closeups! As CCM-equipped products continue to rapidly evolve, these small but powerful optical devices must also keep improving. To ensure that CCMs can produce clear, sharp images within cost and spatial constraints, engineers can analyze their performance with optical ray tracing simulation. Read more
These models are featured in the blog post Analyze Violin Tone and Volume with Multiphysics Modeling. One applies acoustic-structure interaction to study how the air mode resonance is affected by the coupled vibrations in the violin body. The other uses a potential flow approximation to find out how the air flow through the sound holes relates to their shape. Read more
The Hubble Space Telescope (HST) is an example of a standard Cassegrain telescope. The Conic Mirror On Axis 3D part from the COMSOL part library is used to construct the HST Ritchey-Chretien geometry. Multiple release features are used to generate on and off-axis ray and spot diagrams. Download the application files
In mid to late 2020, hotly anticipated 5G-enabled smartphones began to roll out to the general public. One of the key aspects of the new 5G infrastructure that supports these devices is RF filters. These filters, which are used to stop signal interference, can become subject to significant temperature variations resulting in structural deformation, especially in extreme environmental conditions. Engineers designing RF filters for 5G devices must be able to analyze how temperature variations and thermal stresses affect their performance. This is where multiphysics simulation comes into play. Read more
In this blog post, we examine a 12-slot, 10-pole permanent magnet (PM) machine modeled in the COMSOL Multiphysics® software and AC/DC Module. Read more
Thermal management is an important factor in the design and performance of all electromagnetic devices, as it affects device cost, efficiency, reliability, and safety, among other factors. Mathematical modeling and numerical simulation can be used to investigate heat transfer mechanisms, temperature variations, and thermal effects in order to make informed design choices and better predict device performance. Application Note
COMSOL Multiphysics® versie 5.6 biedt snellere solvers waarbij minder geheugen wordt gebruikt, een efficiëntere werkwijze met betrekking tot CAD-geometrieën, Clip Planes, lay-outtemplates voor applicaties en vier nieuwe producten: Fuel Cell & Electrolyzer Module, LiveLink™ for Simulink®, Polymer Flow Module en Liquid & Gas Properties Module. Lees verder
When setting up and running CFD, heat transfer, and acoustics simulations, modeling the material properties correctly is crucial. With the Liquid & Gas Properties Module, you can accurately and easily compute density, viscosity, thermal conductivity, heat capacity, and other properties as functions of composition, pressure, and temperature. Read more
We released COMSOL Multiphysics® version 5.6 on November 11, 2020. Watch this video to get an introduction to the new version, including core functionality updates and news pertaining to specific add-on modules.
Electromagnetic shielding is vital to the power & energy industry because it is used to ensure the operation of technical systems, reduce the unwanted emissions of electromagnetic fields, and more. In this video, you will learn how to model electromagnetic shielding using COMSOL Multiphysics® and the AC/DC Module. Read more and watch the video
Researchers at the Flemish Institute for Technical Research (VITO/EnergyVille) and KU Leuven developed a pseudo-three-dimensional model of a semisolid flow battery to find reliable answers to their design questions, such as how flow rate affects particle discharge and how cell voltage changes during the discharge process. Read more
Batman and Robin. Gin and tonic. Carl Zeiss and Ernst Abbe? Although you may not have heard of that last duo, their collaboration led to advancements in the development of microscopes and optical lenses. This 19th-century story — involving a picturesque German town, microscopes, and a hammer and anvil — also demonstrates what can happen when empirical research and theory come together. Imagine the possibilities when simulation is brought in… Read more
Noninvasive ventilation (NIV) masks supply air to patients in respiratory distress via continuous positive airway pressure (CPAP). Polibrixia, a research company in Italy, set out to design and optimize an oronasal mask for CPAP- and NIV-ventilated patients using multiphysics simulation. Read more
If you are new to simulation, check out the first video, Part 1, “Introduction to COMSOL Multiphysics®”, to see how simulation can benefit optical product design. In Part 2, “COMSOL Applications in Optics and Photonics”, Wallace and Boucher discuss why you should simulate more than just optics. Watch the video
Looking for a quick, guided introduction to electromagnetic coil modeling with the COMSOL Multiphysics® software and the AC/DC Module? We put together a series of video lectures that walk you through the modeling of electromagnetic coils, designed to show the key aspects of building such models. Let’s quickly review what’s in these videos and how to use the series to your best advantage! Read more
Have you ever noticed how your own voice sounds different when you speak while wearing a set of headphones or earbuds? Known as the occlusion effect, this phenomenon also affects the performance and comfort provided by hearing aids, and in turn, their acceptance by hearing-impaired people. Let’s see how the COMSOL Multiphysics® software can help researchers in minimizing this effect to design better hearing aids and earbuds. Read more
This example shows how to use response spectrum analysis to verify the integrity of a structure that is exposed to an earthquake. The building is modeled as a steel frame, using beam elements. Displacements and stresses are computed. Read more
For 2D modeling components, or when using work planes in 3D models, you may have noticed a subtle yet significant change in how geometry can be created in the software. Using Sketch mode, along with constraints and dimensions, you can draw plane geometry and define relations between the geometric entities that you draw. Rest assured, the 2D geometry drawing functionality you have grown accustomed to is still available, but now there are features that enable efficiency like never before. Read more
To balance the broad range of personalized features being built into luxury vehicles, the team of acoustics and simulation experts at HARMAN accounts for different components, acoustics, and unique configurations early on in the design system process, see figure and read more
Computing the spatial derivative of the magnetic field or magnetic flux density is useful in areas such as radiology, magnetophoresis, particle accelerators, and geophysics. One of the most important use cases is the design of magnetic resonance imaging (MRI) machines, where it is necessary to analyze not only the field strength but also the spatial variation of the field. Read more