FINITE ELEMENT ANALYSIS: Solution
by Steve Roensch, President, Roensch & Associates

Third in a four-part series

While the pre-processing and post-processing phases of the finite element method are interactive and time-consuming for the analyst, the solution is often a batch process, and is demanding of computer resource. The governing equations are assembled into matrix form and are solved numerically. The assembly process depends not only on the type of analysis (e.g. static or dynamic), but also on the model’s element types and properties, material properties and boundary conditions.

In the case of a linear static structural analysis, the assembled equation is of the form Kd = r, where K is the system stiffness matrix, d is the nodal degree of freedom (dof) displacement vector, and r is the applied nodal load vector. To appreciate this equation, one must begin with the underlying elasticity theory. The strain-displacement relation may be introduced into the stress-strain relation to express stress in terms of displacement. Under the assumption of compatibility, the differential equations of equilibrium in concert with the boundary conditions then determine a unique displacement field solution, which in turn determines the strain and stress fields. The chances of directly solving these equations are slim to none for anything but the most trivial geometries, hence the need for approximate numerical techniques presents itself.

A finite element mesh is actually a displacement-nodal displacement relation, which, through the element interpolation scheme, determines the displacement anywhere in an element given the values of its nodal dof. Introducing this relation into the strain-displacement relation, we may express strain in terms of the nodal displacement, element interpolation scheme and differential operator matrix. Recalling that the expression for the potential energy of an elastic body includes an integral for strain energy stored (dependent upon the strain field) and integrals for work done by external forces (dependent upon the displacement field), we can therefore express system potential energy in terms of nodal displacement.

Applying the principle of minimum potential energy, we may set the partial derivative of potential energy with respect to the nodal dof vector to zero, resulting in: a summation of element stiffness integrals, multiplied by the nodal displacement vector, equals a summation of load integrals. Each stiffness integral results in an element stiffness matrix, which sum to produce the system stiffness matrix, and the summation of load integrals yields the applied load vector, resulting in Kd = r. In practice, integration rules are applied to elements, loads appear in the r vector, and nodal dof boundary conditions may appear in the d vector or may be partitioned out of the equation.

Solution methods for finite element matrix equations are plentiful. In the case of the linear static Kd = r, inverting K is computationally expensive and numerically unstable. A better technique is Cholesky factorization, a form of Gauss elimination, and a minor variation on the “LDU” factorization theme. The K matrix may be efficiently factored into LDU, where L is lower triangular, D is diagonal, and U is upper triangular, resulting in LDUd = r. Since L and D are easily inverted, and U is upper triangular, d may be determined by back-substitution. Another popular approach is the wavefront method, which assembles and reduces the equations at the same time. Some of the best modern solution methods employ sparse matrix techniques. Because node-to-node stiffnesses are non-zero only for nearby node pairs, the stiffness matrix has a large number of zero entries. This can be exploited to reduce solution time and storage by a factor of 10 or more. Improved solution methods are continually being developed. The key point is that the analyst must understand the solution technique being applied.

Dynamic analysis for too many analysts means normal modes. Knowledge of the natural frequencies and mode shapes of a design may be enough in the case of a single-frequency vibration of an existing product or prototype, with FEA being used to investigate the effects of mass, stiffness and damping modifications. When investigating a future product, or an existing design with multiple modes excited, forced response modeling should be used to apply the expected transient or frequency environment to estimate the displacement and even dynamic stress at each time step.

This discussion has assumed h-code elements, for which the order of the interpolation polynomials is fixed. Another technique, p-code, increases the order iteratively until convergence, with error estimates available after one analysis. Finally, the boundary element method places elements only along the geometrical boundary. These techniques have limitations, but expect to see more of them in the near future.

Next month’s article will discuss the post-processing phase of the finite element method.

copyright 2005 Roensch & Associates. All rights reserved.

Finite Element Analysis: Solution is a post from: UM Life Science Park

When Narcissistic Supply is deficient, the narcissist de-compensates and acts out in a variety of ways. Narcissists often experience psychotic micro-episodes during therapy and when they suffer narcissistic injuries in a life crisis. But can the narcissist “go over the edge”? Do narcissists ever become psychotic?

Some terminology first:

The narrowest definition of psychosis, according to the DSM-IV-TR, is “restricted to delusions or prominent hallucinations, with the hallucinations occurring in the absence of insight into their pathological nature”.

And what are delusions and hallucinations?

A delusion is “a false belief based on incorrect inference about external reality that is firmly sustained despite what almost everyone else believes and despite what constitutes incontrovertible and obvious proof or evidence to the contrary”.

A hallucination is a “sensory perception that has the compelling sense of reality of a true perception but that occurs without external stimulation of the relevant sensory organ”.

Granted, the narcissist’s hold on reality is tenuous (narcissists sometimes fail the reality test). Admittedly, narcissists often seem to believe in their own confabulations. They are unaware of the pathological nature and origin of their self-delusions and are, thus, technically delusional (though they rarely suffer from hallucinations, disorganised speech, or disorganised or catatonic behaviour). In the strictest sense of the word, narcissists appear to be psychotic.

But, actually, they are not. There is a qualitative difference between benign (though well-entrenched) self-deception or even malignant con-artistry and “losing it”.

Pathological narcissism should not be construed as a form of psychosis because:

The narcissists is usually fully aware of the difference between true and false, real and make-belief, the invented and the extant, right and wrong. The narcissist consciously chooses to adopt one version of the events, an aggrandising narrative, a fairy-tale existence, a “what-if” counterfactual life. He is emotionally invested in his personal myth. The narcissist feels better as fiction than as fact but he never loses sight of the fact that it is all just fiction.
Throughout, the narcissist is in full control of his faculties, cognisant of his choices, and goal-orientated. His behaviour is intentional and directional. He is a manipulator and his delusions are in the service of his stratagems. Hence his chameleon-like ability to change guises, his conduct, and his convictions on a dime.
Narcissistic delusions rarely persist in the face of blanket opposition and reams of evidence to the contrary. The narcissist usually tries to convert his social milieu to his point of view. He attempts to condition his nearest and dearest to positively reinforce his delusional False Self. But, if he fails, he modifies his profile on the fly. He “plays it by ear”. His False Self is extemporaneous a perpetual work of art, permanently reconstructed in a reiterative process designed around intricate and complex feedback loops.
Though the narcissistic personality is rigid its content is always in flux. Narcissists forever re-invent themselves, adapt their consumption of Narcissistic Supply to the “marketplace”, attuned to the needs of their “suppliers”. Like the performers that they are, they resonate with their “audience”, giving it what it expects and wants. They are efficient instruments for the extraction and consumption of human reactions.

As a result of this interminable process of fine tuning, narcissists have no loyalties, no values, no doctrines, no beliefs, no affiliations, and no convictions. Their only constraint is their addiction to human attention, positive or negative.

Psychotics, by comparison, are fixated on a certain view of the world and of their place in it. They ignore any and all information that might challenge their delusions. Gradually, they retreat into the inner recesses of their tormented mind and become dysfunctional.

Narcissists can’t afford to shut out the world because they so heavily depend on it for the regulation of their labile sense of self-worth. Owing to this dependence, they are hypersensitive and hypervigilant, alert to every bit of new data. They are continuously busy rearranging their self-delusions to incorporate new information in an ego-syntonic manner.

This is why the Narcissistic Personality Disorder is insufficient grounds for claiming a “diminished capacity” (insanity) defence. Narcissists are never divorced from reality they crave it, and need it, and consume it in order to maintain the precarious balance of their disorganised, borderline-psychotic personality. All narcissists, even the freakiest ones, can tell right from wrong, act with intent, and are in full control of their faculties and actions.

Pathological Narcissism, Psychosis, and Delusions is a post from: UM Life Science Park

Finally, the disputing companies try to resolve their ongoing case through the court. This is known as Commercial Litigation. The companies, like the individual, take the help of an expert attorney to protect their interest. The attorney files the case for a contingent fee. The company involved in litigation, with the support of the attorney, seeks the Litigation Financing Company for its expenses. The company providing Litigation Financing evaluates the case. If the case appears worthy enough, the company offers an advance, often, termed as litigation loan. These are pre-settlement advances of non-recourse nature. That means the company gets its recovery charges if and only if its client company wins the case. But to ensure proper Commercial Litigation Financing one needs to resort to the attorney.

Commercial Litigation Financing is a post from: UM Life Science Park

As any other student of a language would, I bought a Spanish to English/English to Spanish dictionary. Should have been easy enough except for the fact that I have occasionally used words from the dictionary and people have had no idea what I was saying. After consulting with my tutor I discovered my dictionary is a Spanish for Spain dictionary and that was what was causing my problems. Of course this important piece of information was not printed on the dictionary.

Learn Spanish Language Smooth Sailing The Process is a post from: UM Life Science Park

Get a folder (or a three-ring binder), and label it appropriately on the outside cover. There are even folders you can purchase (some schools even make these available to teachers) labeled ‘sub folder’ or ‘emergency plans’. Also make sure you have an appropriate spot for your emergency folder on or in your desk area. Some schools will ask you to keep an emergency plan in the office. In either case, make sure it is easily accessible by a substitute teacher.

Think about keeping class activities to 10 to 15 minute increments. This way the sub will have better control of your kids. Students have difficulties adjusting to changes in their routines, and you don’t want to have to return to discipline referrals.

Keep the information organized and easily accessible for a sub. Remember, the sub won’t know where you normally keep things, and they can’t read your mind. Spell out exactly what you want done, where it can be found, and what you want done with it when they’re finished.

Make sure you have made enough copies of any worksheets so the sub doesn’t have to. And be sure to leave answer keys. Many subs will actually even grade your assignments for you if you ask them in your plans.

Get this done early in the year, and you can save yourself many headaches later, not to mention worries about what will happen in your room if you are unable to be there.

EMERGENCY LESSON PLAN IDEAS:

Language Arts: Include short writing activities involving students opinions. Thus they don’t have to have ‘background’ information, and they can write from their own experiences. Parts of speech review can include mad-libs or easy, fun worksheets.

Math: Leave a calculator activity. These could even be puzzles or partner games. Or give review problems.

Science: Copy a science article and have students read carefully and answer questions. Make speculations and use the scientific method. Or have students create the plans for a lab activity.

Reading: Leave students a copy of a short story or article, and questions to answer. You could even set up a ‘test-taking’ exercise, and discuss appropriate answers and strategies.

Social Studies: Map activities are great for emergency plans. You can even set up a one-day unit on any area/region of the world, including your own town or city.

Everyone gets those situations in life where an emergency has come up, and you don’t have the time (or sometimes the ability) to get a good lesson plan in to school for your students. Maybe you have a family emergency or a disrupted travel plan and you just cannot get into school to leave detailed lessons. That is why it is essential for you to have an emergency lesson plan available and handy.

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Emergency Lesson Plans, Real Lifesaving Tools for Teachers is a post from: UM Life Science Park

Home Schooled adolescents are mainly tutored by one or both parents. The modern school day of eight to three or something in the region can go out the window in some situations. As an alternative, parents may choose to oversee their Childs education every evening after work with the chances of including sections of the weekends. Also, parents can pay for a teacher or make it so that one parent is able to oversee the kids education while the other takes care of earning a wage.

Another point is the curriculum. Home schooling parents must select a program of study that works with, not against, their children’s learning capabilities and styles. You see some kids are visual learners while others need more hands-on assistance by assembling things or taking them to bits. And, the curriculum of your choosing will have to be okayed by the local education board. Check out the different variations that can be located online or that circulate among home schooling groups or appear in your local library. The school superintendent or local press may also be able to help in this area.

It is vitally important to ensure that your children are meeting educational goals and are keeping with their age and grade capabilities. A majority of homeschooled learners actually perform better than their counterparts in conventional schooling, Others can fall behind and may not do well in a one on one setting with the parent. It may well be up to the parents, who may have little or no formal education, to ensure the quality and standard of their kids education.

Last but not least, be sure to supplement at-home education with field trips, community group activities, and perhaps extracurricular participation in sports, music, or art classes. You can get more info about these from the local community institutions, library , or certain county high schools that provide such services. Home-schooled students are among the brightest and best prepared fro college. But take care to make sure that your child receives a quality, top notch education if you decide to choose this option. For further information and advice, contact one of the broad-based home schooling programs or check with the local school board to source other parents in your area who are themselves in the process of homeschool education. If however this turns out to be tough you could easily put an ad in the local press. This way you should be able to find some other homeschooling parents and together the task of homeschool education would hopefully be easier and the end result will be a quality education for your kids.

Homeschool is a Superb Child Education Solution is a post from: UM Life Science Park

Facts & Figures

Panama is a 75,990 square km land area. The official language is Spanish.
However, many Panamanians speak both Spanish and English.

Panamanian weather is pleasantly tropical, and uniform throughout the year.
Nights are usually cool. The average temperature is 27 degrees C.
Panama has two weather seasons: rainy and dry.
The former stretches from March to December.

The time in Panama all year long is the same as the EST (GMT -5).
Panama does not observe Daylight Savings Time.

Panama’s Tourist Attractions

Islands and Beaches

Panama offers an exquisite view and experience of nature at its best. Panama has
Coasts on two oceans: the Pacific and the Caribbean Sea. These coasts, although
Different, are suited for all sorts of activities with the different beaches and
islands found in them. World famous beautiful beaches and islands will make your
visit unforgettable!

Caribbean Sea: On the Caribbean coast, the beaches on the provinces of Colon,
San Blas and Bocas del Toro and their neighbouring islands, are small to
mid-sized and all of them have coral reefs nearby,
They are often near other waterways and most have been formed by coral buildup.
These are the best beaches to do some scuba diving in.

Pacific Ocean: On this coast you’ll find a series of beaches which are easily
Accessible from the Pan-American Road. Many water sports including windsurfing,
surfing and swimming are practiced here. The most popular of these are: Gorgona,
Coronado, San Carlos, El Palmar, Rio Mar, Corona, Sea Cliff, Santa Clara, Playa
Blanca and Farallon.

Panama Canal

Considered one of the Eight Wonders of the World, the Panama Canal is one of
the most fascinating places in the world, where human genius and skill join to
link two oceans and bring the world closer together.

The Panama Canal has a length of approximately 80 kilometers between the
Atlantic and Pacific Oceans. Between 13,000 and 14,000 ships use the Canal
yearly, thanks to the work of approximately 9,000 workers, working 24 hours
a day, 365 days a year, offering transit service to ships of all nations.

The Canal uses a locks system that act as water elevators raising the ships
from sea-level (Atlantic or Pacific) to 26 meters above sea-level (Gatun Lake).
Each set of locks carries the name of the town where it was built: Gatun
(on the Atlantic side), Pedro Miguel and Miraflores (on the Pacific).

The narrowest part of the Canal is Gaillard Cut and it stretches from the Pedro
Miguel Locks to the southern tip of Gatun Lake in Gamboa. This stretch is
approximately 13.7 kilometers long. Take a mini cruise through the Canal and
then check out the Visitor Center in Mira Flores, the ideal place to see the
Canal operating. This installation, recently open to the public, has large
balconies from which visitors can see the locks open and close as the ships
begin or end their transit. Four exhibition rooms, organized by themes, are the
main feature of the Visitor Center. Exhibitions are dedicated to Canal history,
the importance of water as a source of life, the Canal operation and it’s place
in worldwide trading.

Tropical Rainforest

Panama is an ideal country for direct contact with the tropical rainforest.
The only capital in Latin America with a rainforest less than 10 minutes away
from downtown and with easy access to three national parks.

If you would like to go into the tropical rainforest, Panama offers national
parks declared Biosphere Reserves and World Heritage. Rainforests with the
widest variety of flora and fauna in the world (Darien) where we can find over
10,000 plant species and over 1,000 bird species.

Colon Free Zone

In a separated area of 400 hectares the Colon Free Zone is located at the
entrance of the Panama Canal in the Caribbean sector, in the province of Colo
and is considered the second largest of its kind in the world and the first in
the Western Hemisphere. Due to its unique geographic location and easy access
to four major ports, considered the most advanced in Latin America, this is
truly an International Shipping Center.

Also, located next to it, Visitors can find transportation services such as
France Field Airport, a Railroad Terminal, access to the Panama-Colon highway,
cruise terminals and a ground transportation terminal.

The Colon Free Zone is a Global Logistical Center. Its commercial transactions
generate US$ 11,000 million in imports and re-exports. With over 400 hectares
its more than 2,000 companies and more that 250,000 annual visitors reflect the
success of the Colon Free Zone.

Traders and investors can find: The existence for the International Banking
Center, the efficient maritime infrastructure, the largest center for collection
and redistribution in the Americas, the use of the US Dollar as currency,
security both within and around the area, great exhibition rooms,
warehouse and commercial outfits. A great workforce, both specialized and
low cost, a great number of tax incentives and a sophisticated communications
network are just some of the factors that contribute to ease the operations from
the Colon Free Zone and make this the ideal place for International Commerce.

Staying In Panama Made Easy

Getting to Panama is easy with all the transportation modes in this country.
There are land, water and air transportations operating in Panama. Staying in
Panama is also a burden off the shoulders of tourists. There is a wide array of
hotels and accommodations in Panama. These hotels suit any type of budget and
needs. They range from economy to luxury.

Indeed, there are countless reasons why tourists simply love Panama. With all
the attractions and complete comfort, there is no better “home away from home”
destination than Panama.

GO WILD IN PANAMA is a post from: UM Life Science Park

Scholarships are for highly intellectual students only.

Most students are discouraged with the idea that all scholarship grants are reserved for A students. This is 100% untrue. Although many scholarships offered everywhere have something to do with considering the academics performance of students, there are several factors that are also taken into great account. For this matter, scholarship grants can also be available for B and C students.

Many scholarships also have something to do with any field outside the norms of academic performance and grades. For example, there are scholarships for athletes, artists, performers, and so on. There are some scholarships that consider the weight of the applicants involvement in community development. The thing is you dont have to be an Einstein to win a scholarship.

Scholarships are only for the underprivileged

While there are many institutions that provide scholarships for the underprivileged, other scholarships do not fall on this condition. Many scholarships are granted for middle class or even for those who can afford. It is true that most scholarships are based on the need; there are some that arent. As a matter of fact, there are many scholarships that are based on the merits. This means that an applicant is not measured based on the financial capability but on academic proficiency and extracurricular activities.

Scholarships involve money right at the application process

Yes, there are groups or institutions that will do all the work for you. They will find the most suitable scholarship for your need and they will process your scholarship application. Be warned that these might be scams. They do not intent to give you the scholarship you need but the money you earned. If you come across this, never give any payment.

You cannot apply for several types of scholarships

The fact is you can always apply for several types of scholarships you think you are qualified into. Just make sure you give attention on the requirements and you are on the way in getting to a school you always dream of.

College Scholarship Myths You Need To Know is a post from: UM Life Science Park

Well to begin, home schooling and online schools are not the same things although it shares a basic principle, which is that the student studies and learns at home.

These two types of schooling are considered non-traditional learning methods as opposed to the regular schoolroom type of setting. However, being non-traditional is one of the few things these two methods have in common with each other.

Home schooling entails that the child or children, are taught by a parent at home. Although done outside of the classroom setting, children that are home schooled still learn the same things that other children learn in school like basic curriculum.

Also, home schooled children will still go on field trips or other social activities that the parent-teacher will plan, as well as learn music and physical education, among other things. There are many reasons as to why a parent may decide to home school their child or children and some reasons may include the fact that traditional schools have their own philosophy and set of values that they teach to their students, so much so that certain basic lessons will be taught with certain slants toward their own preference. To avoid this, a parent may decide to home school his or her child or children.

Another reason could be that one of the parent’s job entails a lot of traveling and instead of breaking up the family unit, home schooling will allow the flexible of learning anywhere home is. This same flexibility is shared by home schooling with online schools. An online school is available to its enrolled student anywhere, anytime as long as there is a computer connected to the internet.

An online school’s main benefit is the flexibility in time it affords a student. This is why, online schools are preferred by a lot of adults who may want to pursue further studies but cannot do so in a traditional setting because perhaps they already hold jobs or they have other responsibilities that prevent them from learning in a traditional setting. Just like in a regular school, online schools have set curriculums and lesson requirements that an online learner needs to accomplish to earn an online degree.

Home schooling and online schools are considered non-traditional methods of learning only because it is yet to have enough people utilizing them to become the norm. Although they are non traditional, this does not mean that they are any less better than attending a regular traditional school. In fact, in some cases, these non-traditional methods are the best solution available.

A Non-Traditional Tradition is a post from: UM Life Science Park

FINITE ELEMENT ANALYSIS: Pre-processing
by Steve Roensch, President, Roensch & Associates

Second in a four-part series

As discussed last month, finite element analysis is comprised of pre-processing, solution and post-processing phases. The goals of pre-processing are to develop an appropriate finite element mesh, assign suitable material properties, and apply boundary conditions in the form of restraints and loads.

The finite element mesh subdivides the geometry into elements, upon which are found nodes. The nodes, which are really just point locations in space, are generally located at the element corners and perhaps near each midside. For a two-dimensional (2D) analysis, or a three-dimensional (3D) thin shell analysis, the elements are essentially 2D, but may be “warped” slightly to conform to a 3D surface. An example is the thin shell linear quadrilateral; thin shell implies essentially classical shell theory, linear defines the interpolation of mathematical quantities across the element, and quadrilateral describes the geometry. For a 3D solid analysis, the elements have physical thickness in all three dimensions. Common examples include solid linear brick and solid parabolic tetrahedral elements. In addition, there are many special elements, such as axisymmetric elements for situations in which the geometry, material and boundary conditions are all symmetric about an axis.

The model’s degrees of freedom (dof) are assigned at the nodes. Solid elements generally have three translational dof per node. Rotations are accomplished through translations of groups of nodes relative to other nodes. Thin shell elements, on the other hand, have six dof per node: three translations and three rotations. The addition of rotational dof allows for evaluation of quantities through the shell, such as bending stresses due to rotation of one node relative to another. Thus, for structures in which classical thin shell theory is a valid approximation, carrying extra dof at each node bypasses the necessity of modeling the physical thickness. The assignment of nodal dof also depends on the class of analysis. For a thermal analysis, for example, only one temperature dof exists at each node.

Developing the mesh is usually the most time-consuming task in FEA. In the past, node locations were keyed in manually to approximate the geometry. The more modern approach is to develop the mesh directly on the CAD geometry, which will be (1) wireframe, with points and curves representing edges, (2) surfaced, with surfaces defining boundaries, or (3) solid, defining where the material is. Solid geometry is preferred, but often a surfacing package can create a complex blend that a solids package will not handle. As far as geometric detail, an underlying rule of FEA is to “model what is there”, and yet simplifying assumptions simply must be applied to avoid huge models. Analyst experience is of the essence.

The geometry is meshed with a mapping algorithm or an automatic free-meshing algorithm. The first maps a rectangular grid onto a geometric region, which must therefore have the correct number of sides. Mapped meshes can use the accurate and cheap solid linear brick 3D element, but can be very time-consuming, if not impossible, to apply to complex geometries. Free-meshing automatically subdivides meshing regions into elements, with the advantages of fast meshing, easy mesh-size transitioning (for a denser mesh in regions of large gradient), and adaptive capabilities. Disadvantages include generation of huge models, generation of distorted elements, and, in 3D, the use of the rather expensive solid parabolic tetrahedral element. It is always important to check elemental distortion prior to solution. A badly distorted element will cause a matrix singularity, killing the solution. A less distorted element may solve, but can deliver very poor answers. Acceptable levels of distortion are dependent upon the solver being used.

Material properties required vary with the type of solution. A linear statics analysis, for example, will require an elastic modulus, Poisson’s ratio and perhaps a density for each material. Thermal properties are required for a thermal analysis. Examples of restraints are declaring a nodal translation or temperature. Loads include forces, pressures and heat flux. It is preferable to apply boundary conditions to the CAD geometry, with the FEA package transferring them to the underlying model, to allow for simpler application of adaptive and optimization algorithms. It is worth noting that the largest error in the entire process is often in the boundary conditions. Running multiple cases as a sensitivity analysis may be required.

Next month’s article will discuss the solution phase of the finite element method.

copyright 2005 Roensch & Associates. All rights reserved.

Finite Element Analysis: Pre-processing is a post from: UM Life Science Park