| FINITE ELEMENT ANALYSIS:
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| | to express stress in terms of
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| IntroductionFirst in a four-part
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| | displacement. Under the
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| seriesFinite element analysis (FEA) is a
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| | assumption of compatibility, the
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| fairly recent discipline
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| | differential equations of
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| crossing the boundaries of mathematics,
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| | equilibrium in concert with the boundary
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| physics, engineering
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| | conditions then
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| and computer science. The method has
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| | determine a unique displacement field
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| wide application and
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| | solution, which in
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| enjoys extensive utilization in the
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| | turn determines the strain and stress
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| structural, thermal and
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| | fields. The chances
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| fluid analysis areas. The finite
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| | of directly solving these equations are
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| element method is
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| | slim to none for
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| comprised of three major phases:
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| | anything but the most trivial
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| (1) pre-processing, in
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| | geometries, hence the need for
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| which the analyst develops a finite
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| | approximate numerical techniques
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| element mesh to divide
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| | presents itself.A finite element mesh is
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| the subject geometry into subdomains for
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| | actually a displacement-nodal
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| mathematical
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| | displacement relation, which, through
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| analysis, and applies material
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| | the element
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| properties and boundary
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| | interpolation scheme, determines the
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| conditions,
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| | displacement anywhere
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| (2) solution, during which the program
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| | in an element given the values of its
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| derives
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| | nodal dof.
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| the governing matrix equations from the
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| | Introducing this relation into the
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| model and solves for
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| | strain-displacement
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| the primary quantities, and
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| | relation, we may express strain in terms
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| (3) post-processing, in which
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| | of the nodal
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| the analyst checks the validity of the
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| | displacement, element interpolation
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| solution, examines
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| | scheme and differential
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| the values of primary quantities (such
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| | operator matrix. Recalling that the
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| as displacements and
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| | expression for the
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| stresses), and derives and examines
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| | potential energy of an elastic body
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| additional quantities
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| | includes an integral for
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| (such as specialized stresses and error
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| | strain energy stored (dependent upon the
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| indicators).The advantages of FEA are
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| | strain field) and
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| numerous and important. A new
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| | integrals for work done by external
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| design concept may be modeled to
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| | forces (dependent upon
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| determine its real world
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| | the displacement field), we can
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| behavior under various load
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| | therefore express system
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| environments, and may therefore
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| | potential energy in terms of nodal
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| be refined prior to the creation of
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| | displacement.Applying the principle of
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| drawings, when few
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| | minimum potential energy, we may
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| dollars have been committed and changes
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| | set the partial derivative of potential
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| are inexpensive.
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| | energy with respect
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| Once a detailed CAD model has been
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| | to the nodal dof vector to zero,
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| developed, FEA can
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| | resulting in: a summation
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| analyze the design in detail, saving
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| | of element stiffness integrals,
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| time and money by
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| | multiplied by the nodal
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| reducing the number of prototypes
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| | displacement vector, equals a summation
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| required. An existing
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| | of load integrals.
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| product which is experiencing a field
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| | Each stiffness integral results in an
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| problem, or is simply
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| | element stiffness
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| being improved, can be analyzed to speed
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| | matrix, which sum to produce the system
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| an engineering
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| | stiffness matrix,
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| change and reduce its cost. In
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| | and the summation of load integrals
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| addition, FEA can be
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| | yields the applied load
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| performed on increasingly affordable
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| | vector, resulting in Kd = r. In
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| computer workstations
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| | practice, integration rules
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| and personal computers, and professional
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| | are applied to elements, loads appear in
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| assistance is
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| | the r vector, and
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| available.It is also important to
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| | nodal dof boundary conditions may appear
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| recognize the limitations of FEA.
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| | in the d vector or
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| Commercial software packages and the
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| | may be partitioned out of the
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| required hardware,
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| | equation.Solution methods for finite
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| which have seen substantial price
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| | element matrix equations are
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| reductions, still require
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| | plentiful. In the case of the linear
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| a significant investment. The method
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| | static Kd = r,
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| can reduce product
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| | inverting K is computationally expensive
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| testing, but cannot totally replace it.
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| | and numerically
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| Probably most
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| | unstable. A better technique is
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| important, an inexperienced user can
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| | Cholesky factorization, a
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| deliver incorrect
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| | form of Gauss elimination, and a minor
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| answers, upon which expensive decisions
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| | variation on the
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| will be based.
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| | "LDU" factorization theme. The K matrix
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| FEA is a demanding tool, in that the
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| | may be efficiently
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| analyst must be
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| | factored into LDU, where L is lower
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| proficient not only in elasticity or
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| | triangular,
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| fluids, but also in
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| | D is diagonal, and U is
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| mathematics, computer science, and
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| | upper triangular, resulting in LDUd = r.
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| especially the finite
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| | Since L and D are easily inverted,
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| element method itself.Which FEA package
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| | and U is upper
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| to use is a subject that cannot possibly
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| | triangular, d may be determined by
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| be covered in this short discussion, and
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| | back-substitution.
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| the choice involves
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| | Another popular approach is the
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| personal preferences as well as package
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| | wavefront method, which
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| functionality.
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| | assembles and reduces the equations at
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| Where to run the package depends on the
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| | the same time. Some
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| type of analyses
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| | of the best modern solution methods
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| being performed. A typical finite
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| | employ sparse matrix
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| element solution
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| | techniques. Because node-to-node
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| requires a fast, modern disk subsystem
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| | stiffnesses are non-zero
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| for acceptable
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| | only for nearby node pairs, the
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| performance. Memory requirements are of
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| | stiffness matrix has a large
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| course dependent on
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| | number of zero entries. This can be
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| the code, but in the interest of
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| | exploited to reduce
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| performance, the more the
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| | solution time and storage by a factor of
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| better, with 512 Mbytes to 8 Gbytes per
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| | 10 or more.
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| user a representative
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| | Improved solution methods are
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| range. Processing power is the final
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| | continually being developed.
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| link in the
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| | The key point is that the analyst must
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| performance chain, with clock speed,
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| | understand the solution
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| cache, pipelining and
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| | technique being applied.Dynamic analysis
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| multi-processing all contributing to the
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| | for too many analysts means normal modes.
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| bottom line.
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| | Knowledge of the natural frequencies and
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| These analyses can run for hours on the
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| | mode shapes of a
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| fastest
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| | design may be enough in the case of a
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| systems, so computing power is of the
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| | single-frequency
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| essence.One aspect often overlooked when
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| | vibration of an existing product or
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| entering the finite element
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| | prototype, with FEA
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| area is education. Without adequate
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| | being used to investigate the effects of
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| training on the finite
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| | mass, stiffness and
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| element method and the specific FEA
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| | damping modifications. When
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| package, a new user will
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| | investigating a future product,
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| not be productive in a reasonable amount
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| | or an existing design with multiple
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| of time, and may in
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| | modes excited, forced
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| fact fail miserably. Expect to dedicate
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| | response modeling should be used to
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| one to two weeks up
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| | apply the expected
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| front, and another one to two weeks over
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| | transient or frequency environment to
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| the first year, to
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| | estimate the
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| either classroom or self-help education.
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| | displacement and even dynamic stress at
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| It is also
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| | each time step.This discussion has
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| important that the user have a basic
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| | assumed h-code elements, for which the
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| understanding of the
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| | order of the interpolation polynomials
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| computer's operating system.Next month's
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| | is fixed. Another
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| article will go into detail on the
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| | technique, p-code, increases the order
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| pre-processing phase of the finite
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| | iteratively until
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| element method.© 1996-2005 Roensch &
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| | convergence, with error estimates
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| Associates. All rights reserved.
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| | available after one
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| FINITE ELEMENT ANALYSIS:
| |
| | analysis. Finally, the boundary element
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| Pre-processingSecond in a four-part
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| | method places
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| seriesAs discussed last month, finite
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| | elements only along the geometrical
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| element analysis is
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| | boundary. These
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| comprised of pre-processing, solution
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| | techniques have limitations, but expect
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| and post-processing
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| | to see more of them
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| phases. The goals of pre-processing are
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| | in the near future.Next month's article
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| to develop an
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| | will discuss the post-processing phase
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| appropriate finite element mesh, assign
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| | of the finite element method.©
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| suitable material
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| | 1996-2005 Roensch & Associates. All
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| properties, and apply boundary
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| | rights reserved.
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| conditions in the form of
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| | FINITE ELEMENT ANALYSIS:
|
| restraints and loads.The finite element
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| | Post-processingLast in a four-part
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| mesh subdivides the geometry into
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| | seriesAfter a finite element model has
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| elements, upon which are found nodes.
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| | been prepared and checked,
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| The nodes, which are
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| | boundary conditions have been applied,
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| really just point locations in space,
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| | and the model has
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| are generally located
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| | been solved, it is time to investigate
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| at the element corners and perhaps near
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| | the results of the
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| each midside. For a
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| | analysis. This activity is known as the
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| two-dimensional (2D) analysis, or a
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| | post-processing
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| three-dimensional (3D)
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| | phase of the finite element
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| thin shell analysis, the elements are
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| | method.Post-processing begins with a
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| essentially 2D, but
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| | thorough check for problems
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| may be "warped" slightly to conform to a
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| | that may have occurred during solution.
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| 3D surface. An
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| | Most solvers
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| example is the thin shell linear
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| | provide a log file, which should be
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| quadrilateral; thin shell
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| | searched for warnings or
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| implies essentially classical shell
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| | errors, and which will also provide a
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| theory, linear defines
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| | quantitative measure
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| the interpolation of mathematical
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| | of how well-behaved the numerical
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| quantities across the
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| | procedures were during
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| element, and quadrilateral describes the
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| | solution. Next, reaction loads at
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| geometry. For a 3D
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| | restrained nodes should
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| solid analysis, the elements have
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| | be summed and examined as a "sanity
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| physical thickness in all
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| | check". Reaction loads
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| three dimensions. Common examples
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| | that do not closely balance the applied
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| include solid linear
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| | load resultant for a
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| brick and solid parabolic tetrahedral
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| | linear static analysis should cast doubt
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| elements. In
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| | on the validity of
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| addition, there are many special
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| | other results. Error norms such as
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| elements, such as
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| | strain energy density
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| axisymmetric elements for situations in
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| | and stress deviation among adjacent
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| which the geometry,
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| | elements might be looked
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| material and boundary conditions are all
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| | at next, but for h-code analyses these
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| symmetric about an
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| | quantities are best
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| axis.The model's degrees of freedom
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| | used to target subsequent adaptive
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| (dof) are assigned at the
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| | remeshing.Once the solution is verified
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| nodes. Solid elements generally have
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| | to be free of numerical
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| three translational
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| | problems, the quantities of interest may
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| dof per node. Rotations are
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| | be examined. Many
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| accomplished through
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| | display options are available, the
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| translations of groups of nodes relative
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| | choice of which depends
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| to other nodes.
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| | on the mathematical form of the quantity
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| Thin shell elements, on the other hand,
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| | as well as its
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| have six dof per
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| | physical meaning. For example, the
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| node: three translations and three
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| | displacement of a solid
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| rotations. The addition
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| | linear brick element's node is a
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| of rotational dof allows for evaluation
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| | 3-component spatial vector,
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| of quantities
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| | and the model's overall displacement is
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| through the shell, such as bending
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| | often displayed by
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| stresses due to rotation
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| | superposing the deformed shape over the
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| of one node relative to another. Thus,
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| | undeformed shape.
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| for structures in
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| | Dynamic viewing and animation
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| which classical thin shell theory is a
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| | capabilities aid greatly in
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| valid approximation,
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| | obtaining an understanding of the
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| carrying extra dof at each node bypasses
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| | deformation pattern.
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| the necessity of
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| | Stresses, being tensor quantities,
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| modeling the physical thickness. The
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| | currently lack a good
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| assignment of nodal
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| | single visualization technique, and thus
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| dof also depends on the class of
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| | derived stress
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| analysis. For a thermal
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| | quantities are extracted and displayed.
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| analysis, for example, only one
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| | Principal stress
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| temperature dof exists at
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| | vectors may be displayed as color-coded
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| each node.Developing the mesh is usually
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| | arrows, indicating
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| the most time-consuming task
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| | both direction and magnitude. The
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| in FEA. In the past, node locations
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| | magnitude of principal
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| were keyed in manually
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| | stresses or of a scalar failure stress
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| to approximate the geometry. The more
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| | such as the Von Mises
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| modern approach is to
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| | stress may be displayed on the model as
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| develop the mesh directly on the CAD
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| | colored bands. When
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| geometry, which will be
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| | this type of display is treated as a 3D
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| (1) wireframe, with points and curves
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| | object subjected to
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| representing edges,
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| | light sources, the resulting image is
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| (2) surfaced, with surfaces defining
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| | known as a shaded
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| boundaries, or (3)
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| | image stress plot. Displacement
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| solid, defining where the material is.
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| | magnitude may also be
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| Solid geometry is
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| | displayed by colored bands, but this can
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| preferred, but often a surfacing package
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| | lead to
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| can create a
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| | misinterpretation as a stress plot.An
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| complex blend that a solids package will
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| | area of post-processing that is rapidly
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| not handle. As far
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| | gaining
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| as geometric detail, an underlying rule
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| | popularity is that of adaptive
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| of FEA is to "model
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| | remeshing. Error norms such
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| what is there", and yet simplifying
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| | as strain energy density are used to
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| assumptions simply must
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| | remesh the model,
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| be applied to avoid huge models.
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| | placing a denser mesh in regions needing
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| Analyst experience is of
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| | improvement and a
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| the essence.The geometry is meshed with
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| | coarser mesh in areas of overkill.
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| a mapping algorithm or an
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| | Adaptivity requires an
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| automatic free-meshing algorithm. The
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| | associative link between the model and
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| first maps a
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| | the underlying CAD
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| rectangular grid onto a geometric
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| | geometry, and works best if boundary
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| region, which must
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| | conditions may be
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| therefore have the correct number of
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| | applied directly to the geometry, as
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| sides. Mapped meshes
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| | well. Adaptive
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| can use the accurate and cheap solid
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| | remeshing is a recent demonstration of
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| linear brick 3D
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| | the iterative nature
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| element, but can be very time-consuming,
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| | of h-code analysis.Optimization is
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| if not impossible,
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| | another area enjoying recent advancement.
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| to apply to complex geometries.
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| | Based on the values of various results,
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| Free-meshing automatically
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| | the model is
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| subdivides meshing regions into
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| | modified automatically in an attempt to
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| elements, with the
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| | satisfy certain
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| advantages of fast meshing, easy
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| | performance criteria and is solved
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| mesh-size transitioning
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| | again. The process
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| (for a denser mesh in regions of large
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| | iterates until some convergence
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| gradient), and
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| | criterion is met. In its
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| adaptive capabilities. Disadvantages
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| | scalar form, optimization modifies beam
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| include generation of
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| | cross-sectional
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| huge models, generation of distorted
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| | properties, thin shell thicknesses and
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| elements, and, in 3D,
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| | or material
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| the use of the rather expensive solid
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| | properties in an attempt to meet maximum
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| parabolic tetrahedral
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| | stress constraints,
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| element. It is always important to
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| | maximum deflection constraints, and/or
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| check elemental
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| | vibrational frequency
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| distortion prior to solution. A badly
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| | constraints. Shape optimization is more
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| distorted element
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| | complex, with the
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| will cause a matrix singularity, killing
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| | actual 3D model boundaries being
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| the solution. A
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| | modified. This is best
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| less distorted element may solve, but
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| | accomplished by using the driving
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| can deliver very poor
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| | dimensions as optimization
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| answers. Acceptable levels of
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| | parameters, but mesh quality at each
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| distortion are dependent upon
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| | iteration can be a
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| the solver being used.Material
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| | concern.Another direction clearly
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| properties required vary with the type of
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| | visible in the finite element
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| solution.
| |
| | field is the integration of FEA packages
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| A linear statics analysis, for example,
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| | with so-called
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| will require an
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| | "mechanism" packages, which analyze
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| elastic modulus, Poisson's ratio and
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| | motion and forces of
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| perhaps a density for
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| | large-displacement multi-body systems.
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| each material. Thermal properties are
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| | A long-term goal
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| required for a thermal
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| | would be real-time computation and
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| analysis. Examples of restraints are
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| | display of displacements
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| declaring a nodal
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| | and stresses in a multi-body system
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| translation or temperature. Loads
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| | undergoing large
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| include forces, pressures
| |
| | displacement motion, with frictional
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| and heat flux. It is preferable to
| |
| | effects and fluid flow
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| apply boundary
| |
| | taken into account when necessary. It
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| conditions to the CAD geometry, with the
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| | is difficult to
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| FEA package
| |
| | estimate the increase in computing power
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| transferring them to the underlying
| |
| | necessary to
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| model, to allow for
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| | accomplish this feat, but 2 or 3 orders
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| simpler application of adaptive and
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| | of magnitude is
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| optimization algorithms.
| |
| | probably close. Algorithms to integrate
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| It is worth noting that the largest
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| | these fields of
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| error in the entire
| |
| | analysis may be expected to follow the
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| process is often in the boundary
| |
| | computing power
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| conditions. Running
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| | increases.In summary, the finite element
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| multiple cases as a sensitivity analysis
| |
| | method is a relatively recent
|
| may be required.Next month's article will
| |
| | discipline that has quickly become a
|
| discuss the solution phase of the
| |
| | mature method,
|
| finite element method.© 1996-2005
| |
| | especially for structural and thermal
|
| Roensch & Associates. All rights
| |
| | analysis. The costs
|
| reserved.
| |
| | of applying this technology to everyday
|
| FINITE ELEMENT ANALYSIS: SolutionThird
| |
| | design tasks have
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| in a four-part seriesWhile the
| |
| | been dropping, while the capabilities
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| pre-processing and post-processing phases
| |
| | delivered by the
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| of the
| |
| | method expand constantly. With
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| finite element method are interactive
| |
| | education in the technique
|
| and time-consuming for
| |
| | and in the commercial software packages
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| the analyst, the solution is often a
| |
| | becoming more and
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| batch process, and is
| |
| | more available, the question has moved
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| demanding of computer resource. The
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| | from "Why apply FEA?"
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| governing equations are
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| | to "Why not?". The method is fully
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| assembled into matrix form and are
| |
| | capable of delivering
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| solved numerically. The
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| | higher quality products in a shorter
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| assembly process depends not only on the
| |
| | design cycle with a
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| type of analysis
| |
| | reduced chance of field failure,
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| (e.g. static or dynamic), but also on
| |
| | provided it is applied by a
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| the model's element
| |
| | capable analyst. It is also a valid
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| types and properties, material
| |
| | indication of thorough
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| properties and boundary
| |
| | design practices, should an unexpected
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| conditions.In the case of a linear
| |
| | litigation crop up.
|
| static structural analysis, the
| |
| | The time is now for industry to make
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| assembled equation is of the form Kd =
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| | greater use of this and
|
| r, where K is the
| |
| | other analysis techniques.© 1996-2005
|
| system stiffness matrix, d is the nodal
| |
| | Roensch & Associates. All rights
|
| degree of freedom
| |
| | reserved.by Steve Roensch, President,
|
| (dof) displacement vector, and r is the
| |
| | Roensch & AssociatesSteve Roensch is a
|
| applied nodal load
| |
| | mechanical engineering consultant with
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| vector. To appreciate this equation,
| |
| | more than 20 years of professional
|
| one must begin with
| |
| | experience. He has analyzed hundreds of
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| the underlying elasticity theory. The
| |
| | product designs and has served as an
|
| strain-displacement
| |
| | expert witness across many industries,
|
| relation may be introduced into the
| |
| | including giving depositions and court
|
| stress-strain relation
| |
| | testimony.
|
