mixed variational principles and robust finite element implementations of gradient plasticity at small strains summary this work outlines a theoretical and computational framework of gradient plasticity based on a rigorous exploitation of mixed variational principles in contrast to classical local approaches to plasticity based on locally evolving internal variables order parameter fields are taken into account governed by additional balance type pdes including micro structural boundary conditions this incorporates non local plastic effects based on length scales which reflect properties of the material micro structure we develop a unified variational framework based on mixed saddle point principles for the evolution problem of gradient plasticity which is outlined for the simple model problem of von mises plasticity with gradient extended hardening softening response the mixed variational structure includes the hardening softening variable itself as well as its dual driving force the numerical implementation exploits the underlying variational structure yielding a canonical symmetric structure of the monolithic problem it results in a novel finite element fe design of the coupled problem incorporating a long range hardening softening parameter and its dual driving force this allows a straightforward local definition of plastic loading unloading driven by the long range fields providing very robust fe implementations of gradient plasticity this includes a rational method for the definition of elastic plastic boundaries in gradient plasticity along with a post processor that defines the plastic variables in the elastic range we discuss alternative mixed fe designs of the coupled problem including a local global solution strategy of short range and long range fields this includes several new aspects such as extended q1p0 type and mini type finite elements for gradient plasticity all methods are derived in a rigorous format from variational principles numerical benchmarks address advantages and disadvantages of alternative fe designs and provide a guide for the evaluation of simple and robust schemes for variational gradient plasticity copyright 2013 john wiley sons ltd

stochastic modeling of the transform domain varepsilon rm lms algorithm this paper presents a statistical analysis of the transform domain least mean square tdlms algorithm resulting in a more accurate model than those discussed in the current open literature the motivation to analyze such an algorithm comes from the fact that the tdlms presents a higher convergence speed for correlated input signals as compared with other adaptive algorithms possessing a similar computational complexity such a fact makes it a highly competitive alternative to some applications approximate analytical models for the first and second moments of the filter weight vector are obtained the tdlms algorithm has an orthonormal transformation stage accomplishing a decomposition of the input signal into distinct frequency bands in which the interband samples are practically uncorrelated on the other hand the intraband samples are correlated the larger the number of bands the higher their correlation the model is then derived taking into account such a correlation requiring that a high order hyperelliptic integral be computed in addition to the proposed model an approximate procedure for computing high order hyperelliptic integrals is presented a regularization parameter is also considered in the model expressions permitting to assess its impact on the adaptive algorithm behavior an upper bound for the step size control parameter is also obtained through simulation results the accuracy of the proposed model is assessed

alpha trimmed means and their relationship to median filters suppose that x is a finite set of n numbers the trimmed mean of x is obtained by sorting x into ascending order removing trimming a fixed fraction alpha 0 leq alpha leq 0 5 from the high and low ends of the sorted set and computing the average of the remaining values when applied to a sliding window of length lw the trimming process is called trim filtering for 0 5 the trimmed mean of a set is the median of the set and the filtering operation is called median filtering repeated application of a median filter to the output of a previous median filter of the same length lw eventually produces a signal which is invariant to median filtering this final signal is called the root signal this paper explains the relationship between trimmed means and median filters derives a simple straightforward and fast algorithm for applying a median filter and provides a new explanation of the convergence of repeated median filtering to the root signal the latter result incorporates an approach which permits generalization of the associated concepts to a larger class of index map filters

microstatistics in signal decomposition and the optimal filtering problem the author introduces and analyzes a large class of nonlinear filters which are based on signal decomposition and where the estimation in the decomposed space uses linear combinations of either the observation vector the sorted observation vector or in general a nonlinear transformation of the observation vector thus nonlinear filter response characteristics are achieved but with the machinery of linear systems theory available for their optimization and design it is shown that linear filters are a subclass of microstatistic filters and that the optimal linear filter solution is suboptimal in the decomposed signal space the filtering problem reduces to a set of filters operating on the decomposed signals the output is a weighted sum of the decomposed filtered signals the optimal interconnection structure between decomposed signals has complexity o m 1 where m is the cardinality of the decomposition the formulation is given for a radix 1 decomposition and generalized for a radix q decomposition computer simulations illustrate the performance

formulation and numerical exploitation of mixed variational principles for coupled problems of cahn hilliard type and standard diffusion in elastic solids summary this work develops variational principles for the coupled problem of standard and extended cahn hilliard type species diffusion in solids undergoing finite elastic deformations it shows that the coupled problem of diffusion in deforming solids accounting for phenomena like swelling diffusion induced stress generation and possible phase segregation caused by the diffusing species is related to an intrinsic mixed variational principle it determines the rates of deformation and concentration along with the chemical potential where the latter plays the role of a mixed variable the principle characterizes a canonically compact model structure where the three governing equations involved that is the mechanical equilibrium condition the mass balance for the species content and a microforce balance that determines the chemical potential appear as the euler equation of a variational statement the existence of the variational principle underlines an inherent symmetry of the coupled deformation diffusion problem this can be exploited in the numerical implementation by the construction of time discrete and space discrete incremental potentials which fully determine the update problems of typical time stepping procedures the mixed variational principles provide the most fundamental approach to the monolithic finite element solution of the coupled deformation diffusion problem based on low order basis functions they induce in a natural format the choice of symmetric solvers for newton type iterative updates providing a speedup and reduction of data storage when compared with non symmetric implementations this is a strong argument for the use of the developed variational principles in the computational design of deformation diffusion problems copyright 2014 john wiley sons ltd

fast algorithms for training stack filters stack filters constitute a class of nonlinear digital filters possessing a weak superposition property in the paper fast algorithms for training stack filters are introduced the first algorithm is based on a technique previously developed similar to the previous algorithm this improved method requires only simple arithmetic operations increment decrement and local comparison the spl epsiv convergence property of the fast algorithm is established in addition two fast training algorithms are developed for a subclass of stack filters called weighted order statistic filters these fast algorithms are variants of the lms and rls algorithms in linear filters all three fast algorithms developed exploit the repetition property of the training input the factor of speedup is approximately the ratio between the number of quantization levels of the input and the window size

thermodynamically consistent phase field models of fracture variational principles and multi field fe implementations the computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies this can be overcome by a diffusive crack modeling based on the introduction of a crack phase field in this paper we outline a thermodynamically consistent framework for phase field models of crack propagation in elastic solids develop incremental variational principles and consider their numerical implementations by multi field finite element methods we start our investigation with an intuitive and descriptive derivation of a regularized crack surface functional that converges for vanishing length scale parameter to a sharp crack topology functional this functional provides the basis for the definition of suitable convex dissipation functions that govern the evolution of the crack phase field here we propose alternative rate independent and viscous over force models that ensure the local growth of the phase field next we define an energy storage function whose positive tensile part degrades with increasing phase field with these constitutive functionals at hand we derive the coupled balances of quasi static stress equilibrium and gradient type phase field evolution in the solid from the argument of virtual power here we consider a canonical two field setting for rate independent response and a time regularized three field formulation with viscous over force response it is then shown that these balances follow as the euler equations of incremental variational principles that govern the multi field problems these principles make the proposed formulation extremely compact and provide a perfect base for the finite element implementation including features such as the symmetry of the monolithic tangent matrices we demonstrate the performance of the proposed phase field formulations of fracture by means of representative numerical examples copyright 2010 john wiley sons ltd

nonlinear techniques for interference suppression in spread spectrum systems the binary nature of direct sequence signals is exploited to obtain nonlinear filters that outperform the linear filters hitherto used for this purpose the case of a gaussian interferer with known autoregressive parameters is considered using simulations it is shown that an approximate conditional mean acm filter of the masreliez type performs significantly better than the optimum linear kalman bucy filter for the case of interferers with unknown parameters the nature of the nonlinearity in the acm filter is used to obtain an adaptive filtering algorithm that is identical to the linear transversal filter except that the previous prediction errors are transformed nonlinearly before being incorporated into the linear prediction two versions of this filter are considered one in which the filter coefficients are updated using the widrow lms algorithm and another in which the coefficients are updated using an approximate gradient algorithm simulations indicate that the nonlinear filter with lms updates performs substantially better than the linear filter for both narrowband gaussian and single tone interferers whereas the gradient algorithm gives slightly better performance for gaussian interferers but is rather ineffective in suppressing a sinusoidal interferer

wavelet transform domain adaptive fir filtering this paper presents and studies two new wavelet transform domain least mean square lms algorithms the algorithms exploit the special sparse structure of the wavelet transform of wide classes of correlation matrices and their cholesky factors in order to compute a whitening transformation of the input data in the wavelet domain and minimize computational complexity the procedures differ in the exact estimates they use and in the way they identify the data dependent whitening transformation the first approach explicitly computes a sparse estimate of the wavelet domain correlation matrix of the input process it then computes the cholesky factor of that matrix and uses its inverse to whiten the input the complexity of this first approach is o n log sup 2 n in contrast the second approach computes a sparse estimate of the cholesky factor of the wavelet domain correlation matrix of the input process directly this second approach has a computational complexity of o n log n floating point operations however it requires a more complex bookkeeping procedure both algorithms have a convergence rate that is faster than that of time domain lms and discrete fourier transform dft or discrete cosine transform dct based lms procedures the paper compares the two procedures and analyzes their mean and mean square performance

hybrid and enhanced finite element methods for problems of soil consolidation hybrid and enhanced finite element methods with bi linear interpolations for both the solid displacements and the pore fluid pressures are derived based on mixed variational principles for problems of elastic soil consolidation both plane strain and axisymmetric problems are studied it is found that by choosing appropriate interpolation of enhanced strains in the enhanced method and by choosing appropriate interpolations of strains effective stresses and enhanced strains in the hybrid method the oscillations of nodal pore pressures can be eliminated several numerical examples demonstrating the capability and performance of the enhanced and hybrid finite element methods are presented it is also shown that for some situations such as problems involving high poisson s ratio and in other related problems where bending effects are evident the performance of the enhanced and hybrid methods are superior to that of the conventional displacement based method the results from the hybrid method are better than those from the enhanced method for some situations such as problems in which soil permeability is variable or discontinuous within elements since all the element parameters except the nodal displacements and nodal pore pressures are assumed in the element level and can be eliminated by static condensation the implementations of the enhanced method and the hybrid method are basically the same as the conventional displacement based finite element method the present enhanced method and hybrid method can be easily extended to non linear consolidation problems copyright 2006 john wiley sons ltd

quadrilateral h div finite elements we consider the approximation properties of quadrilateral finite element spaces of vector fields defined by the piola transform extending results previously obtained for scalar approximation the finite element spaces are constructed starting with a given finite dimensional space of vector fields on a square reference element which is then transformed to a space of vector fields on each convex quadrilateral element via the piola transform associated to a bilinear isomorphism of the square onto the element for affine isomorphisms a necessary and sufficient condition for approximation of order r 1 in l2 is that each component of the given space of functions on the reference element contain all polynomial functions of total degree at most r in the case of bilinear isomorphisms the situation is more complicated and we give a precise characterization of what is needed for optimal order l2 approximation of the function and of its divergence as applications we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for some standard finite element approximations of h div we also derive new estimates for approximation by quadrilateral raviart thomas elements requiring less regularity and propose a new quadrilateral finite element space which provides optimal order approximation in h div finally we demonstrate the theory with numerical computations of mixed and least squares finite element approximations of the solution of poisson s equation

performance of transform domain lms adaptive digital filters in this paper we analyze the performance particularly the convergence behavior of the transform domain least mean square lms adaptive digital filter adf using the discrete fourier transform and discrete orthogonal transforms such as discrete cosine and sine transforms we first obtain the optimum wiener solution and the minimum mean squared error mse in the transform domain it is shown that the two minimum mse s in the time and transform domains are identical independently of the transforms used we then study the convergence conditions and the steady state excess mse s of the transform domain lms trlms algorithms both for the cases of having a constant and a time varying convergence factors when a constant convergence factor is used the convergence behaviors of the lms and trlms adf s appear to be almost identical provided that each has an appropriate value of the convergence factor depending on the transform used also based on the concept of a self orthogonalizing algorithm in the transform domain it is shown that the convergence speed of the trlms adf can be improved significantly for the same excess mse as that of the lms adf in addition we compare the computational complexities of the lms and trlms adf s finally we investigate by computer simulation the effects of system parameter values and different transforms on the convergence behavior of the trlms adf

transform domain lms algorithm the concept of transform domain adaptive filtering is introduced in certain applications filtering in the transform domain results in great improvements in convergence rate over the conventional time domain adaptive filtering the relationship between several existing frequency domain adaptive filtering algorithms is established applications of the discrete fourier transform dft and the discrete cosine transform dct domain adaptive filtering algorithms in the areas of speech processing and adaptive line enhancers are discussed

a generalization of median filtering using linear combinations of order statistics we consider a class of nonlinear filters whose output is given by a linear combination of the order statistics of the input sequence assuming a constant signal in white noise the coefficients in the linear combination are chosen to minimize the output mse for several noise distributions it is shown that the optimal order statistic filter osf tends toward the median filter as the noise becomes more impulsive the optimal osf is applied to an actual noisy image and is shown to perform well combining properties of both the averaging and median filters a more general design scheme for applications involving nonconstant signals is also given

two families of mixed finite elements for second order elliptic problems two families of mixed finite elements one based on triangles and the other on rectangles are introduced as alternatives to the usual raviart thomas nedelec spaces error estimates inl 2 andh 5 are derived for these elements a hybrid version of the mixed method is also considered and some superconvergence phenomena are discussed

mixed variational potentials and inherent symmetries of the cahn hilliard theory of diffusive phase separation this work shows that the cahn hilliard theory of diffusive phase separation is related to an intrinsic mixed variational principle that determines the rate of concentration and the chemical potential the principle characterizes a canonically compact model structure where the two balances involved for the species content and microforce appear as the euler equations of a variational statement the existence of the variational principle underlines an inherent symmetry in the two field representation of the cahn hilliard theory this can be exploited in the numerical implementation by the construction of time and space discrete incremental potentials which fully determine the update problems of typical time stepping procedures the mixed variational principles provide the most fundamental approach to the finite element solution of the cahn hilliard equation based on low order basis functions leading to monolithic symmetric algebraic systems of iterative update procedures based on a linearization of the nonlinear problem they induce in a natural format the choice of symmetric solvers for newton type iterative updates providing a speed up and reduction of data storage when compared with non symmetric implementations in this sense the potentials developed are believed to be fundamental ingredients to a deeper understanding of the cahn hilliard theory

the use of the modified escalator algorithm to improve the performance of transform domain lms adaptive filters this paper describes a new algorithm that improves the convergence performance of the transform domain least mean square trlms algorithm the algorithm exploits the sparse structure of the correlation matrix of the transformed input process to derive a data dependent gram schmidt orthogonalization type transform of the process we show its faster convergence compared with the time domain least mean square lms algorithm and the dct or the dwt based trlms algorithm the gram schmidt orthogonalization is realized using a modified adaptive escalator algorithm the modification significantly reduces the computational complexity of the adaptive escalator algorithm and determines the computational complexity of the proposed algorithm

transform domain adaptive filters an analytical approach transform domain adaptive filters refer to lms filters whose inputs are preprocessed with a unitary data independent transformation followed by a power normalization stage the transformation is typically chosen to be the discrete fourier transform dft although other transformations such as the cosine transform dct the hartley transform dht or the walsh hadamard transform have also been proposed in the literature the resulting algorithms are generally called dft lms dct lms etc this preprocessing improves the eigenvalue distribution of the input autocorrelation matrix of the lms filter and as a consequence ameliorates its convergence speed in this paper we start with a brief intuitive explanation of transform domain algorithms we then analyze the effects of the preprocessing performed in dft lms and dct lms for first order markov inputs in particular we show that for markov 1 inputs of correlation parameter spl rho spl isin 0 1 the eigenvalue spread after dft and power normalization tends to 1 spl rho l 1 spl rho as the size of the filter gets large whereas after dct and power normalization it reduces to 1 spl rho for comparison the eigenvalue spread before transformation is asymptotically equal to 1 spl rho sup 2 1 spl rho sup 2 the analytical method used in the paper provides additional insight into how the algorithms work and is expected to extend to other input signal classes and other transformations

adaptive stack filtering with application to image processing with the aid of threshold decomposition it is shown that optimal stack filters under the mean absolute error mae criterion are equal to optimal or bayesian classifiers subject to stacking constraints under the mean classification error mce criterion nonadaptive and adaptive constrained least mean absolute lma algorithms are developed for the esti mation of stack filters through the linearization of the unit step function in the objective function the convergence of the al gorithms is proven under certain conditions although the methods do not generally give optimal stack filters under the mae criterion these algorithms have several distinct merits compared to other stack filter optimization methods 1 the op timization problem has a unique solution that approximates the optimal stack filters in the least mean square sense 2 the meth ods can be implemented in the binary and real domains and 3 for stack filters defined by linearly separable positive boolean functions pbf s or weighted order statistic wos filters the number of the parameters to be estimated is reduced to the window width of the filter a comparison between images re stored by the new algorithms and the stack filtering algorithm optimal under the mae criterion confirms the effectiveness of the proposed algorithms

minimum mean absolute error estimation over the class of generalized stack filters a class of sliding window operators called generalized stack filters is developed this class of filters which includes all rank order filters stack filters and digital morphological filters is the set of all filters possessing the threshold decomposition architecture and a consistency property called the stacking property conditions under which these filters possess the weak superposition property known as threshold decomposition are determined an algorithm is provided for determining a generalized stack filter which minimizes the mean absolute error mae between the output of the filter and a desired input signal given noisy observations of that signal the algorithm is a linear program whose complexity depends on the window width of the filter and the number of threshold levels observed by each of the filters in the superposition architecture the results show that choosing the generalized stack filter which minimizes the mae is equivalent to massively parallel threshold crossing decisions making when the decision are consistent with each other

detail preserving ranked order based filters for image processing the theoretical analysis of multistage median filters is developed it is shown that multistage median filters are a combination of max median and min median filters since multistage median filters belong to the class of two dimensional stack filters they have threshold decomposition attributes making their theoretical analysis simple statistical threshold decomposition is applied to derive the statistical characteristics of these filters and the results are used to evaluate the performance of these two types of multistage filters finally a quantitative and qualitative comparison of the multistage filters and of other efficient detail preserving filters is presented the comparisons are made using the mean squared error and the mean absolute error criteria

adaptive stack filtering under the mean absolute error criterion an adaptive filter algorithm is developed for the class of stack filters which is a class of nonlinear filters obeying a weak superposition property the adaptation algorithm can be interpreted as a learning algorithm for a group of decision making units the decisions of which are subject to a set of constraints called the stacking constraints under a rather weak statistical assumption on the training inputs the decision strategy adopted by the group which evolves according to the proposed learning algorithm is shown to converge to an optimal strategy in the sense that it corresponds to an optimal stack filter under the mean absolute error criterion this adaptive algorithm requires only increment decrement and comparison operations and only local interconnections between the learning units implementation of the algorithm in hardware is therefore very feasible an example is provided to show how the adaptive stack filtering algorithm can be used in an application in image processing

stack filters and the mean absolute error criterion a method to determine the stack filter which minimizes the mean absolute error between its output and a desired signal given noisy observations of this desired signal is presented specifically an optimal window width b stack filter can be determined with a linear program with o b2 sup b variables this algorithm is efficient since the number of different inputs to a window width b filter is m sup b if the filter has m valued input and the number of stack filters grows faster than 2 raised to the 2 sup b 2 power it is shown that optimal stack filtering under the mean absolute error criterion is analogous to optimal linear filtering under the mean squared error criterion both linear filters and stack filters are defined by superposition properties both classes are implementable and both have tractable procedures for finding the optimal filter under an appropriate error criterion

fast matlab assembly of fem matrices in 2d and 3d we propose an effective and flexible way to assemble finite element stiffness and mass matrices in matlab we apply this for problems discretized by edge finite elements typical edge finite elements are raviart thomas elements used in discretizations of h div spaces and nedelec elements in discretizations of h curl spaces we explain vectorization ideas and comment on a freely available matlab code which is fast and scalable with respect to time

morphological filters part ii their relations to median order statistic and stack filters this paper extends the theory of median order statistic os and stack filters by using mathematical morphology to analyze them and by relating them to those morphological erosions dilations openings closings and open closings that commute with thresholding the max min representation of os filters is introduced by showing that any median or other os filter is equal to a maximum of erosions moving local minima and also to a minimum of dilations moving local maxima thus os filters can be computed by a closed formula that involves a max min on prespecified sets of numbers and no sorting stack filters are established as the class of filters that are composed exactly of a finite number of max min operations the kernels of median os and stack filters are collections of input signals that uniquely represent these filters due to their translation invariance the max min functional definitions of these nonlinear iliters is shown to be equivalent to a maximum of erosions by minimal with respect to a signal ordering kernel elements and also to a minimum of dilations by minimal kernel elements of dual filters the representation of stack filters based on their minimal kernel elements is proven to be equivalent to their representation based on irreducible sum of products expressions of boolean functions it is also shown that median filtering and its iterations of any signal by convex 1 d windows is bounded below by openings and above by closings a signal is a root fixed point of the median iff it is a root of both an opening and a closing the open closing and clos opening yield median roots in one pass suppress impulse noise similarly to the median can discriminate between positive and negative noise impulses and are computationally less complex than the median some similar results are obtained for 2 d median filtering

a minimization principle for deformation diffusion processes in polymeric hydrogels constitutive modeling and fe implementation abstract this paper presents a recently developed innovative minimization principle for coupled deformation diffusion processes applied to hydrogels and compares this new structure with the classical saddle point formulation in both variational foundation and finite element implementation first balance equations and boundary conditions associated with dissipative fluid transport in solids undergoing large deformation are shown to be rooted in a canonical minimization formulation this two field principle determines the deformation rate and the fluid flux constitutively governed by the scalar free energy and the dissipation potential functions it can be used to derive the well known saddle point formulation by a legendre transformation of the dissipation potential next the variational potential is transformed to its incremental counterpart by means of a discretization in time which offers an intuitive and unconstrained discretization within the finite element method to this end vectorial raviart thomas shape functions are chosen for flux degrees of freedom in order to fulfill the required h d i v b conformity the need for this ansatz space can be interpreted as a counterpart to the lbb condition that arises within the saddle point principle and is usually addressed by mixed element formulations however we are able to demonstrate equivalent or superior performance of the minimization principle in several representative boundary value problems phenomena specific to hydrogels like diffusion induced large volume change with instability patterns in the presence of geometrical constraints are successfully modeled the proposed variational framework can thus be validated and its significance complementary to classical approaches is underlined with the inherent symmetry of the coupled problem as a key feature and consequence of the minimization formulation

fast positive definite linear system solvers the authors show that the m band wavelet transforms of a wide class of covariance matrices consist of subblocks that are essentially banded furthermore they prove that the cholesky factors of the transformed covariance matrices also consist of subblocks that are essentially banded they combine these two observations to construct a fast o n sup 2 algorithm for solving the n spl times n linear positive definite systems of equations that arise in statistical signal processing finally they provide an error analysis of the proposed linear positive definite system solver

nonlinear smoothing filters based on rank estimates of location a class of nonlinear filters that are based on the rank estimates r estimates of location parameters in statistical theory is introduced it is shown how moving window rank filters r filters can be defined starting from rank estimates of location these filters utilize the relative ranks of the observations in each window to produce an output value the idea of rank winsorization is extended to that of averaging only observations which lie within small temporal neighborhoods this leads to a definition of the class of generalized wilcoxon gw filters which are parameterized by three parameters namely the degrees of temporal and rank winsorization and the degree of averaging the gw filters can be defined to have desirable characteristics of edge preservation detail retention and impulse rejection in addition to the property of gaussian noise smoothing performance characteristics of these filters are considered through analysis and simulation the filters show that all three well known classes of robust location estimates the l m and r estimates can be applied to nonlinear smoothing of signals

frequency analysis and synthesis of a class of nonlinear filters nonlinear filters are analyzed from a spectral point of view the equivalent frequency behavior of a filter is described by its linear part which represents the linear nonrecursive filter that best matches in the mean square sense the behavior of the filter the class considered includes all the order statistic filters l filters such as median midrange alpha trim and some new order statistic filters ll filters defined by the authors to generalize l filters and fir filters the formulation leads to a technique for imposing on the nonlinear filters a given spectral behavior a simulation of the design for a low pass ll filter is given