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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.

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It is unavoidable some people are getting deeper into debt. When everything goes badly, they view mortgage lender as an angel who can help to recover from financial difficulty. This is one of alternatives that many people are seeking for and this is a way for them to minimize and consolidate their expenses.

What is a definition of Mortgage? Basically, a mortgage is a legal record or document designed to protect the mortgage lender against delay of payment or the debtor’s refusal to pay the debt.

A mortgage lender can be any financial institution or even an individual who has the capacity to lend money to the borrower. There are, actually, various types of mortgage lenders. The key in selecting a mortgage is to choose the right one that fits your needs. Look for a mortgage that has the capacity to lend you the right amount of money at a reasonable rate of interest. There are 3 places where can lend you money:

1. Bank: The bank is the most common and well-known mortgage lender. You can opt to choose the bank as your mortgage lender for reliability, convenience, and nippy approval on loans. Banks generally work faster in processing your loans as compared to other mortgage lenders. Banks are also a one-stop center for all your lending needs.

2. Mortgage Broker: You can also secure a mortgage through a mortgage broker. A mortgage broker is a type of mortgage lender that usually acts as a middleman and finds the appropriate loan that best fits your needs.

3. Credit Union and Thrifts: You may want to consider credit unions and thrifts as other types of lending institutions where mortgages can be secured.

Whatever type of mortgage lender you choose; your credit history will have a definite influence on the placement of a mortgage and availability of money. Whichever form of mortgage you choose, be sure to do your homework before making a final decision. Get recommendations from friends or relatives who know reliable mortgage lenders. As a final step in the process, be sure to check the mortgage lender’s credentials so you can be certain that your financial transactions will be secure and dependable.

It is wise to pay more attention to this alternative and be careful with it. After all, it’s your money that’s at stake if things will not go on smoothly. So, it would be better to be sure with your mortgage lender even if it means you’re the one who is asking for favor.

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In a generation when physical fitness is given topmost attention, people are always on the lookout for the newest and most effective means for weight loss. Acupuncture, the method of inserting thin, filiform needles on certain points in an individual’s body, has been found to be one effectual method for losing weight.

Not many people may find the idea of being inserted by needles quite comforting in their quest for weight control. However, this ancient Chinese alternative treatment seeks to deliver a control mechanism, enabling the patient to manage hunger cravings more successfully in the long run.

The Skinny on Acupuncture Weight Loss

It has been found that weight gain is directly related to emotions. Other than physical hunger, people reach for a huge chocolate bar or a big platter of burger and fries because of the sense of comfort derived from food and eating. More often than not, excessive weight gain is an emotional issue, rather than a mere physical one. You may not realize it, but you tend to take in more food whenever you’re stressed, upset, or pressured.

In Acupuncture, there are specific spots on the body being targeted by the hair-like needles. These spots are linked to certain areas in the body and by stimulating these spots; the patient will achieve a greater sense of inner balance. As these points are inserted by the needles, certain hormones are released throughout the body. These substances work by helping you manage hunger and efficiently control the impulse to overeat.

How Acupuncture Induces Weight Loss

Most acupuncturists will target the spots behind the ear when it comes to promoting weight loss. When these points behind the ear are stimulated, endorphins are released in large doses. Endorphins are called the feel-good hormones, and these are also often referred to as natural pain and fever relievers. The release of endorphins is what makes you feel better after an increased level of physical activity, as in the case of a good workout.

These bodily compounds allow the patient to experience better relaxation, thus considerably alleviating stress. There is a greater possibility of enhanced weight loss when the patient has already achieved emotional wellness. Endorphins are also released whenever the body experiences low levels of bodily pain, thus their moniker as a natural pain reliever.

Patients of acupuncture typically experience very minimal or no pain at all, however most of the points are located near nerve endings and muscle tissues. As the needles are embedded in the body, signals are sent to the brain thus promoting the release of endorphins from the pituitary gland.

There are also other points in the body that serve as gateways for better weight management. One of these placements promotes a decrease in an individual’s appetite, while another has the ability to reduce water retention in the body. The acupuncturist may choose a multi-targeted approach, depending on the requirements of the patient.

Consulting with a Professional Acupuncturist

There are a number of acupuncture practitioners in existence nowadays, offering a host of services. If you are considering this form treatment for your weight management, make sure that you settle with no less than a trained and professional acupuncturist. A poorly trained acupuncturist may not be able to pinpoint the specific meridian points and worse, may cause unnecessary bodily pain and discomfort.

Moreover, your acupuncturist may require an herbal supplement for your treatment, to encourage a longer-term effect. Your sessions need not be maintained for an extensive period; however you may be required to visit your acupuncturist during the entire course of the sessions.

There are other means for losing weight successfully, and this does not include crash dieting and strenuous work outs. If carried out by trained professionals, acupuncture can be the answer for effective weight management. While it is far from being a cure-all, it may just be the right weight loss solution for you.

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Data recovery or Datenrettung is the process by which the data is recovered from damaged or inaccessible storage media. Data recovery is done from storage media like CD´s, DVD´s, Floppies, Hard disks, Magnetic tapes etc. There are two reasons due to which the data becomes inaccessible. They are logical damage and physical damage to the storage media.

Physical damage to storage media can occur in many ways. Magnetic tapes can break, get crumpled or dirt may settle on the tapes. CD´s and DVD´s can have scratches or the metallic layer may get damaged. Magnetic heads in hard disks can crash or motors may fail. The floppy is notorious for failing frequently due to bending, overheating, cold, dust etc.

There are other reasons like fires, electrical surges etc that can cause data to become inaccessible or lost. There are many methods by which data can be recovered from magnetic media or optical media. The methods most commonly used are Magnetic Force Microscopy (MFM), Scanning Probe Microscopy (SPM), Magnetic Force Scanning Tunneling Microscopy (STM) etc.

In these methods a sharp magnetic tip is placed closed to the surface to be analyzed. It interacts with the stray magnetic field. An image of the data is generated and then repairs are carried out on logical damage and thus the data is recovered. Many companies carry out data recovery.

Logical damage is the damage to the file system. It is not physical but a software problem and also tells us that we need to be cautious and have a quality system for data recovery, datenrettung. It generally occurs due to power cuts, system crashes etc preventing file system structures from being written resulting in file system being left in an inconsistent state.

Logical damage is more common than physical damage. This may cause strange behavior like infinitely recurring file directories, loss of data, system crashes, hard disks reporting negative space etc. The end result is that the operating system cannot mount the file system.

Most operating systems come with repair facilities like, Linux has fcsk utility, Mac has disk utility and Windows has chkdisk facility. There are other specialized programs available which have better repair facilities than the operating systems.

There are also other systems called journaling file systems like NTFS, EFS (used in Windows XP), ext3 and xfs which can be reverted back to their earlier consistent state. These file systems reduce the amount of data loss.

Data back up is the best way to prevent loss of data and it´s the most common system for data recovery, datenrettung. The simplest method is to keep data on drives on which the operating system is not loaded. The other method is to write data on magnetic tapes, CD’s or DVD´s or have online backup.

Backups are very important for databases. Data backup is of 3 types. Full backup means backing up all data. Incremental backup means backing up of only the files that have changed. Differential backup is a mixture of these two. There is another method called continuous data protection in which when data is written to a disk, it is also written to another computer in a network.

Most private persons with a computer do nothing in advance, they just use their computers. Too late, when an incident has happened they learn about data recovery or datenrettung.

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