the motivation for this project is to design a novel optical system for quasi - real time alignment of tracker detector elements used in high energy physics ( hep ) experiments .fox - murphy _ et.al ._ from oxford university reported their design of a frequency scanned interferometer ( fsi ) for precise alignment of the atlas inner detector .given the demonstrated need for improvements in detector performance , we plan to design an enhanced fsi system to be used for the alignment of tracker elements in the next generation of electron positron linear collider detectors .current plans for future detectors require a spatial resolution for signals from a tracker detector , such as a silicon microstrip or silicon drift detector , to be approximately 7 - 10 . to achieve this required spatial resolution, the measurement precision of absolute distance changes of tracker elements in one dimension should be on the order of 1 .simultaneous measurements from hundreds of interferometers will be used to determine the 3-dimensional positions of the tracker elements .the university of michigan group constructed two demonstration frequency scanned interferometer ( fsi ) systems with laser beams transported by air or single - mode optical fiber in the laboratory for initial feasibility studies .absolute distance was determined by counting the interference fringes produced while scanning the laser frequency .the main goal of the demonstration systems was to determine the potential accuracy of absolute distance measurements that could be achieved under controlled conditions .secondary goals included estimating the effects of vibrations and studying error sources crucial to the absolute distance accuracy .the main contents of this proceedings article come from our published paper .however , new material in this paper includes a description of a dual - laser system and a possible optical alignment for a silicon tracker detector .the intensity of any two - beam interferometer can be expressed as where and are the intensities of the two combined beams , and and are the phases . assuming the optical path lengths of the two beams are and , the phase difference in eq .( 1 ) is , where is the optical frequency of the laser beam , and c is the speed of light .  for a fixed path interferometer , as the frequency of the laseris continuously scanned , the optical beams will constructively and destructively interfere , causing `` fringes '' .the number of fringes is where is the optical path difference between the two beams , and is the scanned frequency range .the optical path difference ( opd for absolute distance between beamsplitter and retroreflector ) can be determined by counting interference fringes while scanning the laser frequency .a schematic of the fsi system with a pair of optical fibers is shown in fig.1 .the light source is a new focus velocity 6308 tunable laser ( 665.1 nm 675.2 nm ) . a high - finesse ( ) thorlabs sa200 f- p is used to measure the frequency range scanned by the laser . the free spectral range ( fsr ) of two adjacent f - p peaksis 1.5 ghz , which corresponds to 0.002 nm .a faraday isolator was used to reject light reflected back into the lasing cavity .the laser beam was coupled into a single - mode optical fiber with a fiber coupler .data acquisition is based on a national instruments daq card capable of simultaneously sampling 4 channels at a rate of 5 ms / s / ch with a precision of 12-bits .omega thermistors with a tolerance of 0.02 k and a precision of 0.01 are used to monitor temperature .the apparatus is supported on a damped newport optical table .  in order to reduce air flow and temperature fluctuations ,a transparent plastic box was constructed on top of the optical table .pvc pipes were installed to shield the volume of air surrounding the laser beam . inside the pvc pipes ,the typical standard deviation of 20 temperature measurements was about .temperature fluctuations were suppressed by a factor of approximately 100 by employing the plastic box and pvc pipes .detectors for hep experiments must usually be operated remotely for safety reasons because of intensive radiation , high voltage or strong magnetic fields .in addition , precise tracking elements are typically surrounded by other detector components , making access difficult . for practical hep application of fsi , optical fibers for light delivery and returnare therefore necessary .the beam intensity coupled into the return optical fiber is very weak , requiring ultra - sensitive photodetectors for detection .considering the limited laser beam intensity and the need to split into many beams to serve a set of interferometers , it is vital to increase the geometrical efficiency . to this end , a collimator is built by placing an optical fiber in a ferrule ( 1 mm diameter ) and gluing one end of the optical fiber to a grin lens .the grin lens is a 0.25 pitch lens with 0.46 numerical aperture , 1 mm diameter and 2.58 mm length which is optimized for a wavelength of 630 nm .the density of the outgoing beam from the optical fiber is increased by a factor of approximately 1000 by using a grin lens .the return beams are received by another optical fiber and amplified by a si femtowatt photoreceiver with a gain of .for a fsi system , drifts and vibrations occurring along the optical path during the scan will be magnified by a factor of , where is the average optical frequency of the laser beam and is the scanned frequency range . for the full scan of our laser , .small vibrations and drift errors that have negligible effects for many optical applications may have a significant impact on a fsi system . a single - frequency vibration may be expressed as , where , and are the amplitude , frequency and phase of the vibration , respectively . if is the start time of the scan , eq .( 2 ) can be re - written as /c \end{array } \eqno{(3)}\ ] ] if we approximate , the measured optical path difference may be expressed as \times \\ \sin[\pi f_{vib}(t+t_0)+\phi_{vib } ] \end{array } \eqno{(4)}\ ] ] where is the true optical path difference in the absence of vibrations .if the path - averaged refractive index of ambient air is known , the measured distance is .if the measurement window size is fixed and the window used to measure a set of is sequentially shifted , the effects of the vibration will be evident .we use a set of distance measurements in one scan by successively shifting the fixed - length measurement window one f - p peak forward each time .the arithmetic average of all measured values in one scan is taken to be the measured distance of the scan ( although more sophisticated fitting methods can be used to extract the central value ) . for a large number of distance measurements , the vibration effects can be greatly suppressed .of course , statistical uncertainties from fringe and frequency determination , dominant in our current system , can also be reduced with multiple scans .averaging multiple measurements in one scan , however , provides similar precision improvement to averaging distance measurements from independent scans , and is faster , more efficient , and less susceptible to systematic errors from drift . in this way, we can improve the distance accuracy dramatically if there are no significant drift errors during one scan , caused , for example , by temperature variation .this multiple - distance - measurement technique is called slip measurement window with fixed size , shown in fig.2 .however , there is a trade off in that the thermal drift error is increased with the increase of because of the larger magnification factor for a smaller measurement window size .  in order to extract the amplitude and frequency of the vibration , another multiple - distance - measurement technique called slip measurement window with fixed start pointis used , as shown in fig.2 . in eq .( 3 ) , if is fixed , the measurement window size is enlarged one f - p peak for each shift , an oscillation of a set of measured values indicates the amplitude and frequency of vibration .this technique is not suitable for distance measurement because there always exists an initial bias term , from , which can not be determined accurately in our current system .the typical measurement residual versus the distance measurement number in one scan using the above technique is shown in fig.3(a ) , where the scanning rate was 0.5 nm / s and the sampling rate was 125 ks / s .measured distances minus their average value for 10 sequential scans are plotted versus number of measurements ( ) per scan in fig.3(b ) . the standard deviations ( rms ) of distance measurements for 10 sequential scansare plotted versus number of measurements ( ) per scan in fig.3(c ) .it can be seen that the distance errors decrease with an increase of .the rms of measured distances for 10 sequential scans is 1.6 if there is only one distance measurement per scan ( ) .if and the average value of 1200 distance measurements in each scan is considered as the final measured distance of the scan , the rms of the final measured distances for 10 scans is 41 nm for the distance of 449828.965 , the relative distance measurement precision is 91 ppb .  the standard deviation ( rms ) of measured distances for 10 sequential scans is approximately 1.5 if there is only one distance measurement per scan for closed box data . by using the multiple - distance - measurement technique , the distance measurement precisions for various closed box data with distances ranging from 10 cm to 70 cm collected in the past year are improved significantly ; precisions of approximately 50 nanometers are demonstrated under laboratory conditions , as shown in table 1 .all measured precisions listed in table 1 .are the rms s of measured distances for 10 sequential scans .two fsi demonstration systems , air fsi and optical fiber fsi , are constructed for extensive tests of multiple - distance - measurement technique , air fsi means fsi with the laser beam transported entirely in the ambient atmosphere , optical fiber fsi represents fsi with the laser beam delivered to the interferometer and received back by single - mode optical fibers ..distance measurement precisions for various setups using the multiple - distance - measurement technique .[ cols="^,^,^,^,^ " , ]   based on our studies , the slow fluctuations are reduced to a negligible level by using the plastic box and pvc pipes to suppress temperature fluctuations .the dominant error comes from the uncertainties of the interference fringes number determination ; the fringes uncertainties are uncorrelated for multiple distance measurements . in this case, averaging multiple distance measurements in one scan provides a similar precision improvement to averaging distance measurements from multiple independent scans .but , for open box data , the slow fluctuations are dominant , on the order of few microns in our laboratory .the measurement precisions for single and multiple distance open - box measurements are comparable , which indicates that the slow fluctuations can not be adequately suppressed by using the multiple - distance - measurement technique .a dual - laser fsi system intended to cancel the drift error is currently under study in our laboratory .  in order to test the vibration measurement technique ,a piezoelectric transducer ( pzt ) was employed to produce vibrations of the retroreflector .for instance , the frequency of the controlled vibration source was set to hz with amplitude nanometers .the magnification factors , distance measurement residuals and corrected measurement residuals for 2000 measurements in one scan are shown in fig.3(d ) , fig.3(e ) and fig.3(f ) , respectively .the extracted vibration frequencies and amplitudes using this technique ,  hz , nanometers , agree well with the expectation values .in addition , vibration frequencies at 0.1 , 0.5 , 1.0 , 5 , 10 , 20 , 50 , 100 hz with controlled vibration amplitudes ranging from 9.5 nanometers to 400 nanometers were studied extensively using our current fsi system .the measured vibrations and expected vibrations all agree well within the 10 - 15% level for amplitudes , 1 - 2% for frequencies , where we are limited by uncertainties in the expectations .vibration frequencies far below 0.1 hz can be regarded as slow fluctuations , which can not be suppressed by the above analysis techniques .detailed information about estimation of major error sources for the absolute distance measurement and limitation of our current fsi system is provided elsewhere .a dual - laser fsi system has been built in order to reduce drift error and slow fluctuations occuring during the laser scan .two lasers are operating simultaneously , the two laser beams are coupled into one optical fiber but isolated by using two choppers .the principle of the dual - laser technique is shown in the following .for the first laser , the measured distance , and is drift error during the laser scanning . for the second laser , the measured distance .since the two laser beams travel the same optical path during the same period , the drift errors and should be very comparable . under this assumption, the true distance can be extracted using the formula , where , , the ratio of magnification factors from two lasers .the laser beams are isolated by choppers periodically , so only half the fringes are recorded for each laser , degrading the distance measurement precision . missing fringes during chopped intervals for each laser must be recovered through robust interpolation algorithms .the chopper edge transitions make this interpolation difficult .several techniques are under study .one possible silicon tracker alignment system is shown in fig.4 .the left plot shows lines of sight for alignment in r - z plane of the tracker barrel , the middle plot for alignment in x - y plane of the tracker barrel , the right plot for alignment in the tracker forward region .red lines / dots show the point - to - point distances need to be measured using fsis .there are 752 point - to - point distance measurements in total for the alignment system .more studies are needed to optimize the distance measurments grid .fox - murphy , d.f .howell , r.b .nickerson , a.r .weidberg , `` frequency scanned interferometry(fsi ) : the basis of a survey system for atlas using fast automated remote interferometry '' , nucl .inst . meth .a383 , 229 - 237(1996 )  p.a .coe , d.f .howell , r.b .nickerson , `` frequency scanning interferometry in atlas : remote , multiple , simultaneous and precise distance measurements in a hostile environment '' , meas .technol.15 ( 11 ) : 2175 - 2187 ( 2004 )    hai - jun yang , jason deibel , sven nyberg , keith riles , `` high - precision absolute distance and vibration measurement using frequency scanned interferometry '' , physics/0409110 , to appear in applied optics , july , 2005 .

Abstract: in this paper , we report high - precision absolute distance and vibration measurements performed with frequency scanned interferometry using a pair of single - mode optical fibers . absolute distance was determined by counting the interference fringes produced while scanning the laser frequency . a high - finesse fabry - perot interferometer was used to determine frequency changes during scanning . two multiple - distance - measurement analysis techniques were developed to improve distance precision and to extract the amplitude and frequency of vibrations . under laboratory conditions , measurement precision of 50 nm was achieved for absolute distances ranging from 0.1 meters to 0.7 meters by using the first multiple - distance - measurement technique . the second analysis technique has the capability to measure vibration frequencies ranging from 0.1 hz to 100 hz with amplitude as small as a few nanometers , without a _ priori _ knowledge . a possible optical alignment system for a silicon tracker is also presented .


elastoplastic processes describe behaviour of solid continuum beyond reversible elastic deformations .they are typically described by hysteresis models with a time memory .the rigorous mathematical analysis of elastoplastic problems and the numerical methods for their solution started to appear in the late 70ies and in the early 80ies by the work of c. johnson , h. matthies , v. korneev and u. langer , j. neas and i. hlavek and others .since then a lot of mathematical contributions to computational plasticity have been written , we refer at least to the monographs by j. simo and t. hughes and w. han and b. reddy . in this paper, we focus on the efficient parallel implementation of elastoplastic problems based on the tfeti domain decomposition method .more specifically , we consider an associated elastoplasticity with the von mises plastic criterion and the linear isotropic hardening law ( see e.g. ) .the corresponding elastoplastic constitutive model is discretized by the implicit euler method in time and consequently a nonlinear stress - strain relation is implemented by the return mapping concept ( see e.g. ) .this approach together with the balance equation , the small strain assumption and a combination of the dirichlet and neumann boundary conditions leads to the solution of a nonlinear variational equation with respect to the primal unknown displacement in each time step .such an equation can also be equivalently formulated as a minimization problem with a potential energy functional ( see e.g. ) .  by a finite element space discretization of the one time step problem, we obtain a system of nonlinear equations .the corresponding nonlinear operator is nondifferentiable but strongly semismooth .therefore , it is suitable to choose the semismooth newton method for solving the system since the strong semismoothness together with other properties ensure local quadratic convergence .semismooth functions in finite dimensional spaces and the semismooth newton method were introduced in . in elastoplasticity, the semismoothness was investigated for example in .  in each newton iteration , it is necessary to solve the respective linearized problem . different linear solvers including those based on multigridhave been successfully tested in . moreover , since the linear systems of equations corresponding to the elastic and elastoplastic problems are spectrally equivalent , all preconditioners for elastic problems can be applied to elastoplastic ones as well .a linear solver considered in this paper is based on a feti type domain decomposition method enabling its efficient parallel implementation .the standard feti method ( feti-1 ) was originally introduced by farhat and roux and theoretically analyzed by mandel and tezaur .a systematic overview of using this approach , a body is partitioned into non - overlapping subdomains , an elliptic problem with neumann boundary conditions is defined for each subdomain , and intersubdomain field continuity is enforced via lagrange multipliers .the lagrange multipliers are efficiently solved from a dual problem by a variant of the conjugate gradient algorithm .the first practical implementations exploited only the favorable distribution of the spectrum of the matrix of the smaller problem , known also as the dual schur complement matrix , but such algorithm was efficient only with a small number of subdomains .later , farhat , mandel , and roux introduced a `` natural coarse problem '' whose solution was implemented by auxiliary projectors so that the resulting algorithm became in a sense optimal . here, we use the total - feti ( tfeti ) variant of feti domain decomposition method , where even the dirichlet boundary conditions are enforced by lagrange multipliers .hence all subdomain stiffness matrices are singular with a - priori known kernels which is a great advantage in the numerical solution . with known kernel basiswe can regularize effectively the stiffness matrix without extra fill in and use any standard sparse cholesky type decomposition method for nonsingular matrices .  for systematic overview of domain decomposition methods we refer to .we mention feti - dp and bddc methods as examples of alternatives to the total - feti method .the feti - dp ( dual - primal feti ) is a popular way of avoiding singular stiffness matrices of the local problems and was first introduced by farhat , lesoinne , le tallec , pierson , and rixen . on the other hand , in we gave numerical examples showing that the matrices of the systems arising in tfeti may be better conditioned than those arising in feti - dp .more theoretical and implementation details about feti - dp and its extensions and improvements are given in .the bddc ( balancing domain decomposition by constraints ) was introduced by dohrmann as a simpler primal alternative to the feti - dp .the name of the method was coined by mandel and dohrmann , because it can be understood as further development of the bdd ( balancing domain decomposition ) method .an alternative approach to the solution of elastoplastic problems based on the schwarz domain decomposition method with overlap was introduced in .the structure of the paper is as follows : in section 2 , we introduce a quasistatic scheme of a solid mechanics problem . within this context, we consider both elastic and elastoplastic models .the elastic model is introduced for methodical purposes because it is an essential part of the investigated elastoplastic model .after its time discretization we summarize the resulting one time step problem . in section 3 ,the finite element space discretization of the one time step problem and its nonlinear algebraic formulation are described in details .the semismooth newton method is applied to treat this nonlinearity .the tfeti method combined with the projected conjugate gradient algorithm is derived in section 4 to solve the linearized problems appearing in the newton iterations .finally , the algorithm for solving the whole elastoplastic problem is summarized . in section 5 , the performance of the proposed algorithm is illustrated on numerical experiments .final comments are summarized in section 6 .in this section , we summarize elastic and elastoplastic models in a quasistatic framework .firstly , we introduce basic notation and assumptions that will be used for setting the models .secondly , we describe the linear elastic constitutive model for an isotropic material .thirdly , we introduce the elastoplastic initial value constitutive model based on the described elasticity , the von mises yield function , the associated plastic flow rule and the linear isotropic hardening represented by the accumulated plastic strain .fourthly , we consider time discretization of the constitutive model given by the implicit euler method. the explicit form of the time discretized elastoplastic operator is introduced .let us consider a deformable body occupying a domain with a lipschitz continuous boundary .we will describe the state of the body during a loading process by the cauchy stress tensor , the displacement and the small strain tensor . here is the space of all symmetric second order tensors .other variables that are necessary for defining the elastoplastic models will be introduced in subsection [ subsection_plast_model ] .more details can be found in .the above variables depend on the spatial variable and on the time variable  ] , solve the following system of algebraic equations for the unknowns , , and , subject to the constraints this constitutive problem can be solved explicitly by the return mapping concept ( see e.g. ) .it means that we firstly apply the elastic predictor , i.e. , we verify whether .if it holds then i.e. , the stress increment satisfies the elastic law defined by . if , then by the plastic corrector we have where notice that the second formula in ( [ plast_corr ] ) is correctly defined since the denominator for . let us define the stress and hardening operators , with respect to parameters , , such that for  respectively , where denotes the positive part of the function . then by ( [ trial ] ) , ( [ hard_law_discr ] ) , ( [ elast_pred ] ) , ( [ plast_corr ] ) , ( [ stress_op ] ) and ( [ hardening_op ] ) , for the sake of brevity, we will denote the stress operator with respect to the current parameters and by . by , the operator is potential , lipschitz continuous , strongly monotone , and strongly semismooth on .  let us note that semismoothness was originally introduced by mifflin for functionals .qi and j. sun extended the definition of semismoothness to vector - valued function to investigate the superlinear convergence of the newton method .the strong semismoothness of the lipschitz continuous function means that is directionally differentiable on and has a quadratic approximate property at any , i.e. , for any , , and any , here denotes the set of the clark generalized derivatives of at . herewe will choose the clark generalized derivative of in the following way :  \1 . if , then   \2 .if , then where notice that is not differentiable at , . otherwise .by using ( [ hook_law])([vol - dev_prop ] ) , ( [ stress_op_deriv_elast ] ) and ( [ stress_op_deriv_plast ] ) we can derive the following uniform estimate for ( see e.g. ) :   let us recall that the stress , strain , hardening and displacement variables also depend on a spatial variable .we consider the dependence of on in the following sense : then we can substitute the stress operator , defined by ( [ stress_op ] ) , into the balance equation ( [ weak - balance ] ) to obtain the time discretized elastoplastic problem in the incremental form .[ problem2 ]  _ given the stress field ^{3\times 3}_{sym}eepep$ } } : = { \textit{diag\,}}(1,1,1,2,2,2 ) , \end{array}\ ] ] and respectively . consequently the functions , , , , are defined by , , , , , in the following way :   respectively.  we also introduce the matrix representing the algebraical relation between the strain and the displacement ( the exact form of is in ) ,i.e. , the strain on an element can be found by ( [ displ_restr ] ) in the form   based on the introduced notation we define the non - linear operator , which plays the role of the left hand side in ( [ weak - plast_h ] ) .further we define the tangential and elastic stiffness matrices which represent the bilinear forms and , respectively .in particular , we denote the matrix briefly by , where the reason of will be explained in the next section .let then by using ( [ f_def ] ) , we can rewrite the equation ( [ weak - plast_h ] ) as follows : find such that where is the increment of the load vector .let , , , and denote the restrictions of , , , and given by omitting the entries ( degrees of freedom ) corresponding to the prescribed dirichlet boundary conditions .then we can rewrite the equation ( [ weak - plast_alg ] ) to the following system of non - linear equations : the discretized elastoplastic problem can be solved by the following algorithm :  1.2  initial step : for any find :     the non - linear system of equations ( [ non_system ] ) is solved by the semismooth newton method ( see e.g. ) .the corresponding algorithm is following :  1.2  initialization : find : compute set   here is the relative stopping tolerance and is the restriction of given by omitting the entries ( degrees of freedom ) corresponding to the prescribed dirichlet boundary conditions .since ( [ t_k^o - est ] ) yields the matrices are spectrally equivalent to . by , it means that all types of preconditioners for elastic problems can be applied to the linearized problem in each newton step as well .if , we obtain the elastic - perfectly plastic problem , and become generally singular .such a problem is complicated not only for computations but also for the complete theory in terms of displacements .therefore , the mentioned spectral equivalency seems to be problematic for too small values of .however , most of local stiffness matrices used in assembly of remain in an elastic mode and become more regular in real computations .indeed , we have successfully used the introduced algorithms even for perfectly plastic case .  in , superlinear local convergence of the algorithm has been derived .let us note that the convergence depends on the discretization parameter of the triangulation .therefore we can expect that the finer the mesh , the bigger the number of the newton iterations . in , a damped semismoooth newton method for sucha problem has also been described .such a method has again superlinear local convergence and additionally global convergence .in this section , we describe the tfeti domain decomposition method for solving linearized problems appearing in newton iterations and its optimal solver based on projected conjugate gradient method with preconditioning ( pcgp ) .finally , we summarize tfeti based algorithm for solving the whole elastoplastic problem .   in the previous section , we showed that we need to solve the following system of the linear equations : in each time step and in each newton iteration .since the tfeti domain decomposition method can be described independently of the indices , we will schematically write the problem in the form : where , , are the restriction of with respect to the dirichlet boundary conditions respectively .let us note that is a symmetric and positive semidefinite matrix and is a symmetric and positive definite matrix .therefore the linearized problem ( [ linear_system2 ] ) has a unique solution .the problem ( [ linear_system2 ] ) can be equivalently rewritten as a minimization problem : where   the corresponding functional representation of ( [ linear_system2 ] ) is following : find such that for any , where restricted on is a constant fourth order symmetric and elliptic tensor and restricted on is a constant second order tensor from .we can see that ( [ weak_discr ] ) is related to ( [ linear_system ] ) , if we replace and set and as the functional representation of and , respectively .notice that the problem ( [ weak_discr ] ) has a similar scheme as the elastic problem defined in subsection [ subsection_elast_model ] . therefore the below introduced tfeti method can also be explained for elasticity in the same way .  the variational problem of the type ( [ weak_discr ] )can be equivalently formulated as a minimization problem : where   to apply the tfeti domain decomposition , we tear the body from the part of the boundary with the dirichlet boundary condition , decompose it into subdomains , assign each subdomain by a unique number , and introduce new `` gluing '' conditions on the artificial intersubdomain boundaries and on the boundaries with imposed dirichlet condition ( see figure [ fig2 ] ) .in particular , the polyhedral domain is decomposed into a system of disjoint polyhedral subdomains , .  tfeti domain decomposition with subdomain renumbering , height=207 ]we assume that the decomposition is consistent with the triangulation , i.e. , and define after the decomposition each boundary of consists of three disjoint parts , , and , , where with being the part of which is glued to , .  similarly to the definition of , we can define the spaces , , of piecewise linear and continuous approximations of : ^ 3\;\;\forall t\in \mathcal t_h^p,\;\;v_h^p|_{\gamma^p_u}=0\}.\ ] ] let be a product space and let us note that we slightly distinguish the notation introduced for the tfeti method from the notation introduced in sections [ section.elastoplastic ] and [ section.newton ] . for example , we write for the tfeti method while in section [ section.elastoplastic ] , we used the notation .a similar distinction is also introduced for the below algebraic description of the tfeti method .let be a functional defined on .then the minimization problem ( [ min_form ] ) can be equivalently rewritten into the form : where and solves ( [ min_form ] ) .each function , , can be represented by a vector , , where , is the algebraic representation of and .similarly we can find the vector , , such that is the algebraic representation of the load restricted on and .let the matrix represent the gluing conditions introduced in ( [ def_k_h ] ) and the dirichlet boundary conditions introduced in ( [ def_v_h_p ] ) .both matrices can be combined into one constraint matrix , \quad \mathbf{b } \in \mathbb{r}^{{\tt m}\times { \tt n}},\quad { \tt m } = { \tt m}_g + { \tt m}_u.\ ] ] typically is much smaller than .let us note that can be assembled to have different forms : redundant , non - redundant or orthonormal .the rows of in the standard redundant form are vectors of the order with zero entries except 1 and -1 at appropriate positions .the orthonormal form is obviously obtained from the redundant one by only special treating of the rows of corresponding to the nodes shared by more subdomains , i.e. , for each of such nodes we take all corresponding rows in the redundant form , othonormalize them , and remove dependent ones .this can be done in parallel . for more detailssee  .in fact all forms are applicable but due to simplicity of our presentation we use the orthonormal form of .  let the matrix , denotes a symmetric positive semidefinite block diagonal matrix , where here is the algebraic representation of and is a restriction operator for a displacement vector to a local element .the diagonal blocks , , which correspond to the subdomains , are positive semidefinite sparse matrices with known kernels , the rigid body modes .the algebraical formulation of ( [ min_form2 ] ) is following :   even though ( [ equ.3radky ] ) is a standard convex quadratic programming problem , its formulation is not suitable for numerical solution .the reasons are that is typically ill - conditioned , singular , and very large .  the complications mentioned above may be essentially reduced by applying the duality theory of convex programming ( see , e.g. , dostl ) , where all the constraints are enforced by the lagrange multipliers .the lagrangian associated with problem ( [ equ.3radky ] ) is it is well known that ( [ equ.3radky ] ) is equivalent to the saddle point problem : in sense that solves if and only if solves .the solution of leads to the equivalent problem to find satisfying : the system is uniquely solvable which is guaranteed by the following necessary and sufficient conditions  : notice that is the condition on the full row - rank of .let us mention that an orthonormal basis of is known a - priori and that its vectors are columns of , .the first equation in is satisfied if and for an appropriate and arbitrary matrix satisfying .here is a generalized inverse matrix whose application on a vector can be efficiently implemented ( see remark [ rem_pseudo ] ) .[ rem_pseudo ] the diagonal block , , which corresponds to the subdomain , is positive semidefinite sparse matrix with known kernel basis created by the rigid body modes .the first note about practical implementation of the action of the generalized inverse based on modified cholesky factorization can be found in .this approach does not solve how to detect carefully zero pivots .in contrast to we use the kernel basis and fixing nodes strategy to effectively regularize all blocks without extra fill in and then decompose them using any standard sparse cholesky type factorization method for nonsingular matrices . the action of on a vector is naturally parallelized with respect to the subdomains and computed using backward and forward substitutions .[ rem_pseudo_standa ] notice that the pseudoinverse can also be applied on vectors which do not belong to .therefore we can write in the other text . the random error corresponding with this partitionis neglicted with respect to the implementation of introduced in remark [ rem_pseudo ] .the condition can be equivalently written as further substituting into the second equation in we arrive at summarizing and we find that the pair satisfies : where , , , and   since is of full row - rank as follow from , the inverse exists and is well defined and represents the orthogonal projector onto . applying on the first equation in and checking that we eliminate and obtain that satisfies : in practical computations , we further decompose into two orthogonal components and , substitute them in to and get the problem with , where is sparse and can be treated efficiently by parallel direct solvers such as mumps .equation is solved efficiently by the projected conjugate gradient method with preconditioning ( pcgp ) . in numerical examples ,we consider both the computationally less expensive lumped preconditioner to in the form and the ( quasi- ) optimal dirichlet preconditioner in the form introduced in , where is the schur complement to the block of corresponding to the interior nodes and is the restriction of to the remaining nodes , i.e. , to the nodes with imposed dirichlet and gluing conditions .the condition number estimates contain a linear factor of ( is the domain decomosition step and is the discretization step ) for the lumped preconditioner as opposed to a polylogarithmic bound for the dirichlet preconditioner .more details about other forms of the dirichlet preconditioners and estimates of condition number can be found in .we obtain the following algorithmic scheme for the solution of :  [ algorithm_tfeti ]  1.2  set , , , and .compute .set .compute from by pcgp : ,  until convergence * do * project precondition re - project    * then stop .* . set .compute .compute .action of on a vector may be efficiently implemented by the sparse cholesky factorization of .   in this subsection, we summarize the above proposed algorithms for solving elastoplastic problems and modify them with respect to the use of the tfeti domain decomposition method .[ tfeti_elasto_plastic_algorithm ]  1.2   ( initial step ) set ( zero approximation ) restrict into subdomain variables , call algorithm [ algorithm.compute_variables ] with ( , ) to find output variables , , , , .collect into global variables ,  ( cycle over subdomains ) solve by algorithm [ algorithm_tfeti ] the problem : such that  ( displacement update ) * then stop * ( newton iter . )  ( cycle over time steps )  the assembling procedure for subdomain data looks as follows .[ algorithm.compute_variables ]  1.2  input : . compute ( volume of the element )   ( cycle over elements ) output :   loop over subdomains and all subdomain operations may be implemented in parallel .parallelization of feti / tfeti is based on distributing matrix portions among processing units .this allows algorithms to be almost the same in sequential and parallel versions ; only data structure implementation differs .most of computations ( subdomain operations ) appearing are purely local and therefore parallelizable without any data transfers .each of cores works with the local part associated with its subdomains .natural effort using the massively parallel computers is to maximize the number of subdomains so that the sizes of the subdomain stiffness matrices are reduced .this accelerates their factorization and subsequent pseudoinverse application which belongs to the most time consuming action . on the other hand , negative effect ofthat is an increase of the null space dimension and the number of lagrange multipliers .this leads to larger coarse problems , i.e. , applications of the projector which are scalable only up to a few thousands of cores and then the coarse problem solution starts to dominate . for the numerical solution of such large problems we recommend to use a hybrid feti method proposed algorithms were implemented in matsol library developed in matlab and parallelized using matlab distributed computing server and matlab parallel toolbox . for all computations we use maximum 28 cores with 2 gb memory per core of the hp blade system , model blc7000 .the numbers of subdomains are chosen to keep the number of nodes per subdomain approximately constant except for the coarsest mesh level .the performance is demonstrated on an elastoplastic homogeneous thin plate of sizes with the circular hole of radius 1 in the center .its geometry with imposed boundary conditions and indicated symmetry planes is depicted in figure [ fig04 g ] .thus we consider only eighth of the thin plate in our computations ( see figure [ fig04g2 ] ) and impose symmetry conditions on three symmetry surfaces . a similar benchmark was solved in .the elastoplastic body is made of homegeneous isotropic material with the parameters where and are the young modulus and the poisson ratio , respectively , which are related with the lam coefficients and by the following formulas : the indicated traction forces with the history of loading taking into account are prescribed by the function ,\ t_0=0,\ t^*=\frac{1}{4}.\ ] ] since the elastoplastic model is rate - independent and any local unloading is not expected with respect to the prescribed load history , the results should be independent of the chosen time discretization .let us consider two variants of the equidistant time discretization characterized by the time step :  * , , * , .for the spatial discretization of , let us consider five levels of tetrahedral meshes generated by ansys with nodes decomposed into subdomains using metis .an example of such decomposition is depicted in figure [ fig04d ] and a zoom near the hole in figure [ fig04d2 ] .      in table[ tab : diferent_meshes ] , we report on numerical results for the time discretization , different mesh levels , the dirichlet and the lumped preconditioners .the stopping tolerances of the newton and the pcgp algorithms are respectively .we see that the number of the newton iterations remains almost constant for all meshes .similar behavior was observed in the numerical results by .note that even the total number of the pcgp iterations ( which means the sum of pcgp iterations over all newton steps ) increases only moderately for finer meshes and both preconditioners .we observe similarly as in  that the dirichlet preconditioner is worse because of medium size of our problems , i.e. , the total number of pcgp iterations is less than for the lumped preconditioner but not sufficiently .  in table[ tab : martin5_upr ] , the number of the pcgp iterations , the number of plastic elements , and the relative convergence criterion are reported for each newton s iteration of algorithm [ tfeti_elasto_plastic_algorithm ] . in this case , the time discretization , the finest mesh level , and the lumped preconditioner are considered together with sharper stopping tolerances so that more newton s iterations can be studied .since the initial newton iteration is taken as zero vector ( of displacements ) , the first iteration of the newton method actually solves a purely elastic problem .the second and further iterations already reflect an elastoplastic behaviour with stabilizing number of plastic elements and higher number of pcgp iterations .this is caused by larger condition number of a considered matrix in accordance with estimate .once plastic elements are identified in the 7th step , we observe the quadratic convergence of the newton method .such behavior agrees with the theoretical results in .  in table[ tab : time_steps ] , the computational history of algorithm [ tfeti_elasto_plastic_algorithm ] for the time discretization is documented .we consider the finest mesh level , the stopping tolerances ( [ tolerances ] ) and the lumped preconditioner only . the corresponding development of the plastic zone is depicted for the times , , , and in figures [ fig11 ] - [ fig14 ] , respectively .the growing zone of plastic elements results from the monotonically increased loading .distributions of the von mises stress and the total displacement at the final time are depicted in figures [ fig15 ] and [ fig16 ] , respectively .mesh level & 1 & 2 & 3 & 4 & 5 + number of mesh nodes & 520 & 1 623 & 9 947 & 166 374 & 309 546 + number of mesh elements & 1 441 & 6 279 & 48 287 & 931 709 & 1 758 907 +  + number of subdomains & 1 & 1 & 8 & 135 & 270 + number of cpu cores & 2 & 2 & 9 & 28 & 28 + number of primal variables & 1 560 & 4 869 & 33 414 & 623 904 & 1 183 101 + number of dual variables & 284 & 618 & 5 448 & 136 904 & 272 053 +  + number of plastic elements & 252 & 2 152 & 20 790 & 420 760 & 796 520 + number of newton iterations .& 5 & 6 & 6 & 6 & 6 + total number of pcgp iterations & 237 & 391 & 573 & 718 & 724 + total time in seconds & 8 & 60 & 77 & 928 & 1796 +  + number of plastic elements & 252 & 2 152 & 20 790 & 420 759 & 796 517 + number of newton iterations .& 5 & 6 & 6 & 6 & 6 + total number of pcgp iterations & 235 & 339 & 440 & 482 & 486 + total time in seconds & 9 & 174 & 207 & 1532 & 3274 +  .computational history of algorithm [ tfeti_elasto_plastic_algorithm ] for the time discretization and the finest mesh level [ cols="^,^,^ , > , > , > " , ]           +      comparing time discretizations and we see that the resulting number of plastic elements differs at the final time . the difference is less than and is caused by the roundoff errors , the use of iterative solvers , and the numerical evaluation of the yield function which decides whether an element plasticizes or not .we have proposed the algorithm for the efficient parallel implementation of elastoplastic problems with the von mises plastic criterion and the linear isotropic hardening law which is based on the tfeti domain decomposition method . for the time discretization we used implicit euler method and for the space discretization of the respective one time step elastoplastic problem the finite element method . the latter results in a system of nonlinear equations with a strongly semismooth and strongly monotone operator .thus the semismooth newton method was applied and respective linearized problems were solved in parallel using tfeti .the performance of our algorithm was demonstrated on the 3d elastoplastic thin plate with the hole in the center and prescribed loading history .numerical results for different time discretizations and mesh levels were presented and discussed . local quadratic convergence of the semismooth newton method was observed after identification of the plastic zone which is in accordance with the theoretical results .our future plan is to apply the proposed tfeti based algorithm on elastoplastic multi - body contact problems of mechanics .this has already been done successfully for the pure elastic case .this work was supported by the european regional development fund in the it4innovations centre of excellence project ( cz.1.05/1.1.00/02.0070 ) , the project spomech reg .cz.1.07/2.3.00/20.0070 , and postdoci ii reg .cz.1.07/2.3.00/30.0055 within operational programme education for competitiveness .the first , the third and the fourth author acknowledge the support of the project 13 - 18652s ( ga cr ) and the second 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part ii - numerical realization _ , in proceedings of the xii international conference on computational plasticity - fundamentals and applications ( eds . e. onate , d.r.j .owen , d. peric , b. suarez ) , barcelona , spain , 999 - 1009 , 2013 .ermk , m. , kozubek , t. , markopoulos , a. , _ an efficient feti based solver for elasto - plastic problems of mechanics _ , computational plasticity xi - fundamentals and applications , complas xi , 13301341 , 2011 .dostl , z. , kozubek , t. , markopoulos , a. , brzobohat , t. , vondrk , v. , horyl , p. , _ theoretically supported scalable tfeti algorithm for the solution of multibody 3d contact problems with friction _ , comput .205 , 110120 , 2012 .dostl , z. , kozubek , t. , vondrk , v. , brzobohat , t. , markopoulos , a. , _ scalable tfeti algorithm for the solution of multibody contact problems of elasticity _ , int .methods eng .82 , 13841405 , 2010 .farhat , c. , lesoinne , m. , letallec , p. , pierson , k. , rixen , d. , _ feti - dp : a 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Abstract: we propose an algorithm for the efficient parallel implementation of elastoplastic problems with hardening based on the so - called tfeti ( total finite element tearing and interconnecting ) domain decomposition method . we consider an associated elastoplastic model with the von mises plastic criterion and the linear isotropic hardening law . such a model is discretized by the implicit euler method in time and the consequent one time step elastoplastic problem by the finite element method in space . the latter results in a system of nonlinear equations with a strongly semismooth and strongly monotone operator . the semismooth newton method is applied to solve this nonlinear system . corresponding linearized problems arising in the newton iterations are solved in parallel by the above mentioned tfeti domain decomposition method . the proposed tfeti based algorithm was implemented in matlab parallel environment and its performance was illustrated on a 3d elastoplastic benchmark . numerical results for different time discretizations and mesh levels are presented and discussed and a local quadratic convergence of the semismooth newton method is observed .