The mesh generation

The mesh generation Describe general methods (structured, unstructured, hybrid, adaptive, etc.) and discuss their key features and applications A key step of the finite element method for numerical computation is mesh generation. One is given a domain (such as a polygon or polyhedron; more realistic versions of the problem allow curved domain boundaries) and must partition it into simple â€Å"elements† meeting in well-defined ways. There should be few elements, but some portions of the domain may need small elements so that the computation is more accurate there. All elements should be â€Å"well shaped† (which means different things in different situations, but generally involves bounds on the angles or aspect ratio of the elements). One distinguishes â€Å"structured† and â€Å"unstructured† meshes by the way the elements meet; a structured mesh is one in which the elements have the topology of a regular grid. Structured meshes are typically easier to compute with (saving a constant factor in runtime) but may require more elements or worse-shaped elements. Unstructured meshes are often computed using quadtrees, or by Delaunay triangulation of point sets; however there are quite varied approaches for selecting the points to be triangulated The simplest algorithms directly compute nodal placement from some given function. These algorithms are referred to as algebraic algorithms. Many of the algorithms for the generation of structured meshes are descendents of â€Å"numerical grid generation† algorithms, in which a differential equation is solved to determine the nodal placement of the grid. In many cases, the system solved is an elliptic system, so these methods are often referred to as elliptic methods. It is difficult make general statements about unstructured mesh generation algorithms because the most prominent methods are very different in nature. The most popular family of algorithms is those based upon Delaunay triangulation, but other methods, such as quadtree/octree approaches are also used. Delaunay Methods Many of the commonly used unstructured mesh generation techniques are based upon the properties of the Delaunay triangulation and its dual, the Voronoi diagram. Given a set of points in a plane, a Delaunay triangulation of these points is the set of triangles such that no point is inside the circumcircle of a triangle. The triangulation is unique if no three points are on the same line and no four points are on the same circle. A similar definition holds for higher dimensions, with tetrahedral replacing triangles in 3D. Quadtree/Octree Methods Mesh adaptation, often referred to as Adaptive Mesh Refinement (AMR), refers to the modification of an existing mesh so as to accurately capture flow features. Generally, the goal of these modifications is to improve resolution of flow features without excessive increase in computational effort. We shall discuss in brief on some of the concepts important in mesh adaptation. Mesh adaptation strategies can usually be classified as one of three general types: r-refinement, h-refinement, or p-refinement. Combinations of these are also possible, for example hp-refinement and hr-refinement. We summarise these types of refinement below. r-refinement is the modification of mesh resolution without changing the number of nodes or cells present in a mesh or the connectivity of a mesh. The increase in resolution is made by moving the grid points into regions of activity, which results in a greater clustering of points in those regions. The movement of the nodes can be controlled in various ways. On common technique is to treat the mesh as if it is an elastic solid and solve a system equations (suject to some forcing) that deforms the original mesh. Care must be taken, however, that no problems due to excessive grid skewness arise. h-refinement is the modification of mesh resolution by changing the mesh connectivity. Depending upon the technique used, this may not result in a change in the overall number of grid cells or grid points. The simplest strategy for this type of refinement subdivides cells, while more complex procedures may insert or remove nodes (or cells) to change the overall mesh topology. In the subdivision case, every â€Å"parent cell† is divided into â€Å"child cells†. The choice of which cells are to be divided is addressed below. For every parent cell, a new point is added on each face. For 2-D quadrilaterals, a new point is added at the cell centroid also. On joining these points, we get 4 new â€Å"child cells†. Thus, every quad parent gives rise to four new offsprings. The advantage of such a procedure is that the overall mesh topology remains the same (with the child cells taking the place of the parent cell in the connectivity arrangement). The subdivision process is similar for a triangular parent cell, as shown below. It is easy to see that the subdivision process increases both the number of points and the number of cells A very popular tool in Finite Element Modelling (FEM) rather than in Finite Volume Modelling (FVM), it achieves increased resolution by increasing the order of accuracy of the polynomial in each element (or cell). In AMR, the selction of â€Å"parent cells† to be divided is made on the basis of regions where there is appreciable flow activity. It is well known that in compressible flows, the major features would include Shocks, Boundary Layers and Shear Layers, Vortex flows, Mach Stem , Expansion fans and the like. It can also be seen that each feature has some â€Å"physical signature† that can be numerically exploited. For eg. shocks always involve a density/pressure jump and can be detected by their gradients, whereas boundary layers are always associated with rotationality and hence can be dtected using curl of velocity. In compressible flows, the velocity divergence, which is a measure of compressiblity is also a good choice for shocks and expansions. These sensing paramters which can indicate regions of flow where there are activity are referred to as ERROR INDICATORS and are very popular in AMR for CFD. Just as refinement is possible by ERROR INDICATORS as mentioned above, certain other issues also assume relevance. Error Indicators do detect regions for refinement, they do not actually tell if the resolution is good enough at any given time. In fact the issue is very severe for shocks, the smaller the cell, the higher the gradient and the indicator would keep on picking the region, unless a threshold value is provided. Further, many users make use of conservative values while refining a domain and generally end up in refining more than the essential portion of the grid, though not the complete domain. These refined regions are unneccesary and are in strictest sense, contribute to unneccesary computational effort. It is at this juncture, that reliable and resonable measure of cell error become necessary to do the process of â€Å"coarsening†, which would reduce the above-said unnecessary refinement, with a view towards generatin an â€Å"optimal mesh†. The measures are given by sensors referred to as ERROR ESTIMATORS, literature on which is in abandunce in FEM, though these are very rare in FVM. Control of the refinement and/or coarsening via the error indicators is often undertaken by using either the solution gradient or soultion curvature. Hence the refinement variable coupled with the refinement method and its limits all need to be considered when applying mesh adaptation A hybrid model contains two or more subsurface layers of hexahedral elements. Tetrahedral elements fill the interior. The transition between subsurface hexahedral and interior tetrahedral elements is made using degenerate hexahedral (pyramid) elements. High quality stress results demand high quality elements, i.e., aspect ratios and internal angles as close to 1:1 and 90 °, respectively, as possible. High quality elements are particularly important at the surface. To accommodate features within a component, the quality of elements at the surface of a hexahedral model generally suffers, e.g., they are skewed. Mating components, when node-to-node contact is desired, can also adversely affect the models element quality. Even more difficult is producing a tetrahedral model that contains high quality subsurface elements. In a hybrid model, the hexahedral elements are only affected by the surface mesh, so creating high quality elements is easy. Minimal effort is required to convert CAD data into surface grids using the automated processes of pro-surf. These surface grids are read by pro-am. The surface grid is used to extrude the subsurface hexahedral elements. The thickness of each extruded element is controlled so that high quality elements are generated. The interior is filled automatically with tetrahedral elements. The pyramid elements that make the transition are also generated automatically. A hybrid model will generally contain many more elements than an all-hexahedral model thus increasing analysis run-time. However, the time saved in the model construction phase the more labor intensive phase more than makes up for the increased run-time. Overall project time is reduced considerably. Also, as computing power increases, this â€Å"disadvantage† will eventually disappear. Hexahedral Meshing ANSYS Meshing provides multiple methods to generate a pure hex or hex dominant mesh. Depending on the model complexity, desired mesh quality and type, and how much time a user is able to spend meshing, a user has a scalable solution to generate a quick automatic hex or hex dominant mesh, or a highly controlled hex mesh for optimal solution efficiency and accuracy. Mesh Methods: Automated Sweep meshing Sweepable bodies are automatically detected and meshed with hex mesh when possible Edge increment assignment and side matching/mapping is done automatically Sweep paths found automatically for all regions/bodies in a multibody part Defined inflation is swept through connected swept bodies User can add sizing controls, mapped controls , and select source faces to modify and take control over the automated sweeping Adding/Modifying geometry slices/decomposition to the model also greatly aids in the automation of getting a pure hex mesh. Thin Solid Sweep meshing This mesh method quickly generates a hex mesh for thin solid parts that have multiple faces as source and target. Can be used in conjunction with other mesh methods User can add sizing controls, mapped controls, and select source faces to modify and take control over the automated sweeping MultiZone Sweep meshing This advanced sweeping approach uses automated topology decomposition behind the scenes to attempt to automatically create a pure hex or mostly hex mesh on complicated geometries Decomposed topology is meshed with a mapped mesh or a swept mesh if possible. A user has the option to allow for free mesh in sub-topologies that cant be mapped or swept. Supports multiple source/target selection Defined inflation is swept through connected swept bodies User can add sizing controls, mapped controls and select source faces to modify and take control over the automated meshing Hex-dominant meshing This mesh method uses an unstructured meshing approach to generate a quad dominant surface mesh and then fill it with a hex dominant mesh This approach generally gives nice hex elements on the boundary of a chunky part with a hybrid hex, prism, pyramid, test mesh internally Tetrahedral Meshing The combination of robust and automated surface, inflation and tet meshing using default physics controls to ensure a high-quality mesh suitable for the defined simulation allows for push-button meshing. Local control for sizing, matching, mapping, virtual topology, pinch and other controls provide additional flexibility, if needed. Mesh Methods: Patch conforming mesh method: Bottom-up approach (creates surface mesh, then volume mesh) Multiple triangular surface meshing algorithms are employed behind the scenes to ensure a high quality surface mesh is generated, the first time From that inflation layers can be grown using several techniques The remaining volume is meshed with a Delaunay-Advancing Front approach which combines the speed of a Delaunay approach with the smooth-transitioned mesh of an advancing front approach Throughout this meshing process are advanced size functions that maintain control over the refinement, smoothness and quality of the mesh Patch independent mesh method: Top-down approach (creates volume mesh and extracts surface mesh from boundaries) Many common problems with meshing occur from bad geometry, if the bad geometry is used as the basis to create the surface mesh, the mesh will often be bad (bad quality, connectivity, etc.) The patch independent method uses the geometry only to associate the boundary faces of the mesh to the regions of interest thereby ignoring gaps, overlaps and other issues that give other meshing tools countless problems. Inflation is done as a post step into the volume mesh. Since the volume mesh already exists, collisions and other common problems for inflation are known ahead of time. Note: For volume meshing, a tetrahedral mesh generally provides a more automatic solution with the ability to add mesh controls to improve the accuracy in critical regions. On the contrary, a hexahedral mesh generally provides a more accurate solution, but is more difficult to generate. Shell and Beam Meshing For 2-D planar (axisymmetric), shell and beam models, ANSYS Meshing provides efficient tools for quickly generating a high quality mesh to accurately simplify the physics. Mesh Methods for shell models: Default surface meshing Multiple surface meshing engines are used behind the scenes to provide a robust, automated surface mesh consisting of all quad, quad dominant or all tri surface mesh. User can add sizing controls, and mapped controls to modify and take control over the automated meshing Uniform surface meshing Orthogonal, uniform meshing algorithm that attempts to force an all quad or quad dominant surface mesh that ignores small features to provide optimum control over the edge length Describe key features of ALL existing meshing options in Ansys Mesh module and discuss their applications The meshing tools in ANSYS Workbench were designed to follow some guiding principles: Parametric: Parameters drive system Persistent: Model updates passed through system Highly-automated: Baseline simulation w/limited input Flexible: Able to add additional control w/out complicating the workflow Physics aware: Key off physics to automate modelling and simulation throughout system Adaptive architecture: Open system that can be tied to a customers process CAD neutral, meshing neutral, solver neutral, etc. By integrating best in class meshing technology into a simulation driven workflow, ANSYS Meshing provides a next generation meshing solution.

India as a Land of Infinite Variety Essay

â€Å"So far as I am able to judge, nothing has been left undone, either by man or nature, to make India the most extraordinary country that the sun visits on his rounds. Nothing seems to have been forgotten, nothing overlooked. India is the cradle of the human race, the birthplace of human speech, the mother of history, the grandmother of legend, and the great grandmother of tradition.† Mark Twain, the writer of this quotation, was obviously impressed by the subcontinent. Travel agents say it’s nearly impossible not to be impressed, astonished and amazed by India—all within the first hour off the plane—for this nation is blessed with sacred sites and ancient ruins, each with a mesmerizing story to tell, creating a treasure-filled world of travel opportunities. Towering Mountains, Vast Deserts and Lush Rainforests Nestled between Pakistan, China and Nepal, India is bordered in the north by the world’s highest mountain chain, while further south, tropical rain forests mingle with windswept deserts. To complete the geographic smorgasbord, palm-fringed beaches trim the peninsula like ribbon on a huge Christmas gift. The only thing missing is a glacier, but no one seems to mind. Visitors and locals alike do not complain about India’s astonishing cultural diversity either, for the nation boasts an inspired, imaginative burst of religions and cultures, races and tongues, resulting from 5,000 years of history. India’s long, historical significance and great natural wealth have lured a long succession of foreign influences, traders and craftsmen, each having left their distinctive imprint on the country. Whether you come to rejuvenate spiritually, relax on the countless beaches or live like a queen in the land of princes, India’s amazing diversity offers everything you could ever want in a holiday. With so much to choose from, it’s wise to enlist the aid of a trusted travel agent. Paragons of advice, travel agents suggest that, because of India’s size and diversity, it’s best to focus your journey around a particular tour or theme. This allows you to experience in depth one tantalizing aspect of the country while still enjoying the variety that makes India a top travel destination. The Golden Triangle: New Delhi, Agra, Jaipur Of India’s many tours, the Golden Triangle is one of the most popular, for the three cities that form the triangle—New Delhi, Agra and Jaipur—feature the images that epitomize India. Inside this triangle you’ll see a panorama of majestic architectural creations that feed the area’s rich traditions, like the symphony of marble that is the Taj Mahal (arguably the world’s most perfect structure), the imperial elegance of New Delhi and the desert city of Jaipur. | Taj Mahal| Jaipur is the capital of Rajasthan, one of the largest of India’s 28 states and host to the exotic Desert Adventure tour. Rajasthan is known as the Land of Kings, where sumptuous palaces dot the desert landscape and battle-scarred fortresses stand tall atop hills like sentinels. You’ll begin your adventure at Jodhpur, an ancient stronghold on the edge of the Thar Desert, before traveling along trade routes to the unforgettable golden fortress at Jaisalmer and beyond. Beyond India’s present-day heritage is a past steeped in spiritual lore, enlightening travelers who tour the Buddhist Circuit. Walk in the footsteps of Siddhartha Gautama—the gentle sage who became the Buddha—and sit beneath the Holy Bodhi tree at Bodhgaya, under which he gained enlightenment in 528 BC. Visit Sarnath, where the Buddha gave the world his first sermon, and Kushinagar, the site of his death. This tour is a spiritual journey that will stay with you for a lifetime. ‘Th e Land of Coconuts’ Another Indian journey that will surely stick with you for a lifetime is a cruise through the Backwaters of Kerala. Kerala, on the southwestern tip of the peninsula, is called the land of coconuts, where palm trees shade nearly the entire state from the tropical sun. All along its coast, exotic backwaters, canals, lagoons and inlets create an intricate maze stretching nearly 1,200 miles throughout the land. Traveling by boat along the emerald-tinted backwaters is a magical experience. Any trip you make to incredible India will be a magical experience, especially if you book your journey through your friendly, neighborhood travel agent. Travel agents can combine any of the above tours or extend them to include exciting nights in bustling cities like Calcutta, New Delhi and Mumbai (Bombay). They can send you along other fabulous tours of India, to the Temple Trail of the south, across the Seven Sisters of the northeast or through the jungles that inspired Kipling in the Heart of Heritage. Always warm and inviting, India is a land of timeless beauty with 5,000 years of life under its ornate belt.Contact a trusted travel agent today and start exploring the infinite variety of this magical land.

Ibm International Business Machines Essay

International Business Machines (IBM) is the world’s top provider of computer hardware. It has 80 years of leadership in helping business innovate. IBM alliance with the companies of all sizes around the world and compete almost in every market worldwide. The company makes desktop and notebook PCs, mainframe and servers, storage systems, and peripherals, among its thousands of products. IBM is also one of the largest providers of both software and semiconductors. With the help of the partnership, the company become leading providers of the e-business solution world widely designed specific solution to meet the needs of companies of all sizes. According to IBM ‘s mission statement, the marketplace is on the top of IBM’s principles. â€Å"The marketplace is the driving force behind everything we do.† The second principle is related to technology. â€Å"At our core, we are a technology company with an overriding commitment to quality.† With these missions, the company had an alliance with dozen of high-tech companies worldwide. The company made its constant effort to keep up with the fast-moving competitors. In the year 2002, IBM reorganized its hardware business; the company had merged its desktop and laptop operations. IBM utilizes the services of both Sanmina-SCI and Solectron by acquired manufacturing facilities from IBM. It had also formed a joint venture with Hitachi to combine its disk drive operation. In the software business, IBM is the second largest provider after Microsoft. The acquisitions of Lotus Development and Tivoli in the early move enabled IBM to become a pioneer in server operating system software. IBM is expanding its software operation to e-commerce infrastructure by purchasing the database operations of Informix (2001) and application integration products from CrossWorlds Software (2002). It also has plans for the acquisition of development tool maker Rational Software. In the third principle of IBM’s mission statement, it is stated that the company’s primary measures of success are customer satisfaction and shareholder’s value. The company had made consistent progress in meeting customer satisfaction. In 1998, IBM had established the E-Team that provides public agencies and corporations with the most highly evolved enterprise-level collaborative software. E-team enables customers to prepare better, respond faster, and recover sooner whenever health and safety, the environment, and public or business assets are at risk. Recently, the company had announced the delivery of the first industry standard interface for the IBM Enterprise storage Server. It is designed to help customers efficiently manage storage systems in a multi-vendor storage network. In late January, the expanding relationship between IBM and Siebel Systems was designed to help the business improve the quality of their customer transactions while reducing the cost of ownership. In committing to improve the value of shareholder, the IBM board of directors recently declared a regular quarterly cash dividend of $.15 per common share. It is to be recorded on February 10 and payable on March 10. In the company’s philosophy, the basic value is stated in the next principle: â€Å"We operate as an entrepreneurial organization with a minimum of bureaucracy and a never-ending focus on productivity. The company’s vision is included in the 5th and 6th principle: â€Å"We never lose sight of our strategic vision, we think and act with a sense of urgency.† In recent times, IBM, the information technology group, had just implemented its strategic vision for the competitive advantage in its market place. Which is announcing the launch of its engineering and technology services business unit in Europe. The unit, part of IBM’s global services division, will offer technology-orientated outsourcing and consulting services to a range of industrial clients in such sectors as aerospace, automotive and defense. This unit is steered by a 100-strong team of engineers and scientists based in Mainz, Germany, will be responsible for the European, Middle Eastern and African markets. In the last two principles, the company believes that its employees are the valuable assets. â€Å"Outstanding, dedicated people make it all happen, particularly when we work together as a team.† â€Å"We are sensitive to the needs of all employees and to the communities in which we operate. IBM’s successful is composition with the outstanding persons. Many of IBM former employees had become successful entrepreneur. Coastal Federal, the state’s second-largest credit union with $1.3 billion in assets and 12 branches statewide. Coastal Federal is a great example. Founded 35 years ago by former IBM employees, Coastal Federal may be one of the few lending institutions statewide that could deploy this technology without criticism. About 48,000 of Coastal’s 118,000 members are past or present IBM employees (and their relatives). Many of them are computer programmers and software developers accustomed to dealing with remote delivery systems. A branch with remote tellers usually employs one-half to one-third the number of employees that a conventional branch does, according to Diebold. Six people work at the Coastal Federal branch at North Pointe, compared with 12 to 17 people at one of the credit union’s regular branches. It took Sharon Lyons, a retired IBM employee and Coastal Federal customer, fewer than three minutes to deposit a check with the remote teller. Marc Quinn, Vice President of American Technology is another great example of outstanding former IBM employee. He joined AmTech from C.E. Unterberg Towbin, where he spent the past two years as a senior equity salesman and was recognized as one of the firm’s top producers. He started his career on Wall Street with Kaufman Bros., applying his 15 years of technology experience with IBM. At IBM, Marc was recognized as one of the top salesman within the Finance and Insurance Industry Verticals. IBM is always imaged as one of the world’s leading companies in terms of societal and environmental concerns. According to the recent news, IBM, United Devices and Accelrys will team with leading researchers and the Department of Defense on a Gild computing project aimed at finding a cure for smallpox. In addition, IBM will also help The Chemicals Sector Cyber-Security Information Sharing Forum to speed development and execution of industry benchmarks for cyber-security. In comparison with the Johnson and Johnson credo, IBM set the priority to the marketplace while Johnson and Johnson’s first credo is customer. IBM’s second principle is emphasis in technology. In the other hand, Johnson and Johnson highlight in supplies. IBM weight the customer satisfaction same as shareholder’s value. In contrast, Johnson and Johnson weight them in the two ends. The only common principle that they treated the same is the environment. By reading the articles in the newspaper and the internal resources, IBM had made the constant effort and continous improvement in meeting the principles on its mission statement. IBM had become a worldwide successful high technology company. Its hardware and software operation is ever expanding. IBM had always innovate new strategic to assist its customers in cutting cost and improving the quality of transaction. The company declared the dividend quarterly to its shareholders. The company also playing an important role in social and environment concern by assist the Department of Defense in finding a cure for smallpox and give help in speed development for cyber-security. IBM is very successful in serving its hardware and software services industry worldwide.

Earn an Ivy League Degree Online

Almost all of the eight ivy league universities offer some form of online courses, certificates, or degree programs. Find out how you can get a top-notch online education from Brown, Columbia, Cornell, Dartmouth, Harvard, Princeton, UPenn, or Yale. Brown Brown offers two blended (online plus face-to-face) degree programs. The  IE-Brown Executive MBA program  offers professionals the chance to get a global education over a 15-month period. MBA students work together online and have five week-long sessions in person. The in-person meetings are in Madrid, Spain; Brown University in Providence, United States; and Cape Town, Africa. The Executive Master of Healthcare Leadership degree is an accelerated program for healthcare professionals. The 16-month program requires online students to meet on campus between the start and end of each term – four times total. Brown also offers online pre-college courses for advanced learners in grades 9-12. Subjects such as â€Å"So, You Want to be a Doctor?† and â€Å"Writing for College and Beyond,† prepare students for their upcoming college experience. Columbia Through the Teacher’s College, Columbia offers online certificates in â€Å"Cognition and Technology,† â€Å"Designing Interactive Multimedia Instruction,† and â€Å"Teaching and Learning with Technology.† Students can also enroll in one of two fully online education Masters degrees. The Computing in Education M.A. helps educational professionals prepare to work with technology in schools. The Diabetes Education and Management M.S. prepares health care workers to educate and advocate for improved understanding about diabetes. The Columbia Video Network enables students to earn advanced engineering degrees from home. Virtual students have no residency requirements and have the same access to their professors as traditional students. Degrees available online include M.S. in Computer Science, M.S. in Electrical Engineering, M.S. in Engineering and Management Systems, M.S. in Materials Science, M.S. in Mechanical Engineering, P.D. in Computer Science, P.D. in Electrical Engineering, P.D. in Mechanical Engineering. Students may also take individual online courses in medicine and religion through Columbia’s online programs. Cornell Through the  eCornell program, students can take individual courses and earn certificates completely online. Multi-course certificate programs are available in fields such as Finance and Managerial Accounting, Healthcare, Hospitality and Food Services Management, Human Resources Management, Leadership and Strategic Management, Management Essentials, Marketing, Sales Leadership, Product Leadership and Systems Design, and Plant-Based Nutrition. eCornell courses are designed and taught by Cornell faculty. They have set start and end dates, but are taught asynchronously. Courses and certificates offer students continuing education credits. Dartmouth Dartmouth College has a very limited number of online options. Students can earn The Dartmouth Institute (TDI) Certificate in The Fundamentals of Value-Based Healthcare by completing six online courses. The courses are generally not available for those outside of the certificate program. Health professionals are required to view a limited number of one-hour live streaming sessions, which are usually conducted on Wednesdays. Presenters lecture on topics such as Health Care Finance, Shared Decision Making in Patient-Centered Care, Health Care Informatics, and Understanding the Implications of Variation. Harvard Through the Harvard Extension School, students can take individual online courses, earn certificates, or even earn a degree. The  Bachelor of Liberal Arts degree program allows students to earn an undergraduate degree with the guidance of top-notch professors. Potential students â€Å"earn their way in† by earning a grade of B or higher in three introductory courses. Students must complete four courses on campus, but the rest of the degree can be completed through online options. Degree candidates have access to a variety of Harvard resources including internships, seminars, and research assistance. The Master of Liberal Arts in Extension Studies in the  field of finance or general management degree can be earned by taking 12 courses. Four of these courses must be traditional or blended courses. For distance learning students, blended courses can be taken by traveling to campus for one-weekend session per course. Additional blended Masters programs are available in Psychology, Anthropology, Biology, English, and more. Most require some evening courses on campus. Graduate certificates can be earned fully online and enrollment is open (no application required). Harvard Extension certificates can be earned in the fields of management, sustainability and environmental management, science and information technology, and social sciences. Notable certificates include Business Communication, Cybersecurity, Nonprofit Management, Marketing Management, Green Building and Sustainability, Data Science, Nanotechnology, Legal Studies, and  Software Engineering. Princeton Sorry, online learners. Princeton isn’t offering any courses or degree programs completely online at this time. UPenn While the University of Pennsylvania doesn’t offer any fully online degrees or certificates, the Penn Online Learning Initiative allows students to take  individual courses. Online courses are offered in Arts and Sciences, Executive Education, Nursing, Dentistry, and also English Language Test Preparation. Generally, students interested in these courses will need to apply to the university as a visiting student. Yale Every year, Yale students enroll in virtual courses through Yale Summer Online. Current students or graduates from other colleges are also invited to enroll in these for-credit courses. The course sessions are five-weeks long, and students are required to participate in a weekly live video group meeting with faculty. Some of the class offerings include: Abnormal Psychology, Econometrics and Data Analysis I, Milton, Modern American Drama and Moralities of Everyday Life.