doubling and projection a method of constructing two level designs of resolution iv given a two level regular fractional factorial design of resolution iv the method of doubling produces another design of resolution iv which doubles both the run size and the number of factors of the initial design on the other hand the projection of a design of resolution iv onto a subset of factors is of resolution iv or higher recent work in the literature of projective geometry essentially determines the structures of all regular designs of resolution iv with n n 4 1 in terms of doubling and projection where n is the run size and n is the number of factors these results imply that for instance all regular designs of resolution iv with 5n 16 n n 2 must be projections of the regular design of resolution iv with n 2 factors we show that for 9n 32 n 5 n 16 all minimum aberration designs are projections of the design with 5n 16 factors which is constructed by repeatedly doubling the 2 5 1 design defined by i abcde to prove this result we also derive some properties of doubling including an identity that relates the wordlength pattern of a design to that of its double and a result that does the same for the alias patterns of two factor interactions

three dimensional element configurations for the discontinuous enrichment method for acoustics hexahedral and tetrahedral elements are proposed for the discontinuous enrichment method dem for boundary value problems governed by the helmholtz equation a procedure for obtaining sets of approximately uniform spherical enrichment directions with high flexibility in the choice of their number is constructed conditioning considerations indicate preferred representations of the oscillatory basis functions dispersion properties are used to rate the performance of different element configurations numerical tests assess accuracy indicating that high order dem elements exhibit little dispersion the dispersion and the numerical results are in a good agreement the proposed configurations of dem elements become more competitive as the number of enrichments and lagrange multipliers is increased copyright 2008 john wiley sons ltd

high order multiscale discontinuous galerkin methods for the one dimensional stationary schrodinger equation abstract in this paper we extend our previous work on the second order multiscale discontinuous galerkin dg method for one dimensional stationary schrodinger equations with oscillating solutions dong et al 2016 to high order we propose two types of high order multiscale finite element spaces and prove that the resulting dg methods with these spaces converge optimally with respect to the mesh size h in l 2 norm when h is small enough numerically we observe that these two multiscale dg methods achieve at least the second order convergence without any resonance errors when h e and optimal high order convergence when h e where e is the scale of the wave length we also demonstrate their ability to capture highly oscillating solutions of the schrodinger equation in the application of the resonant tunneling diode rtd model

a new category of hermitian upwind schemes for computational acoustics ii two dimensional aeroacoustics this is the second paper of a series in which we build and study third order hermitian upwind schemes for numerically solving linear aeroacoustics in this paper we extend the method to non stationary flows for this purpose we employ a family of six wave models to discretize the space operator this family is parametrized by an acoustic propagation angle from this wave modelling specific boundary conditions are proposed to treat effectively subsonic supersonic boundary conditions a sequence of numerical simulations is then carried out and makes it possible to examine the effectiveness of the scheme

constructing tables of minimum aberration pn m designs fries and hunter 1980 presented a practical algorithm for selecting standard 2 n m fractional factorial designs based on a criterion they called minimum aberration in this article some simple results are presented that enable the fries hunter algorithm to be used for a wider range of n and m and for designs with factors at p levels where p 2 is prime examples of minimum aberration 2 n m designs with resolution r 4 are given for m n m 9 a matrix is given for generating 3 n m designs with m n m 6 which have or nearly have minimum aberration

spectral methods for hyperbolic problems we review the current state of fourier and chebyshev collocation methods for the solution of hyperbolic problems with an eye to basic questions of accuracy and stability of the numerical approximations throughout the discussion we emphasize recent developments in the area such as spectral penalty methods the use of filters the resolution of the gibbs phenomenon and issues related to the solution of nonlinear conservations laws such as conservation and convergence we also include a brief discussion on the formulation of multi domain methods for hyperbolic problems and conclude with a few examples of the application of pseudospectral collocation methods for solving nontrivial systems of conservation laws

a comparison of high order polynomial and wave based methods for helmholtz problems the application of computational modelling to wave propagation problems is hindered by the dispersion error introduced by the discretisation two common strategies to address this issue are to use high order polynomial shape functions e g hp fem or to use physics based or trefftz methods where the shape functions are local solutions of the problem typically plane waves both strategies have been actively developed over the past decades and both have demonstrated their benefits compared to conventional finite element methods but they have yet to be compared in this paper a high order polynomial method p fem with lobatto polynomials and the wave based discontinuous galerkin method are compared for two dimensional helmholtz problems a number of different benchmark problems are used to perform a detailed and systematic assessment of the relative merits of these two methods in terms of interpolation properties performance and conditioning it is generally assumed that a wave based method naturally provides better accuracy compared to polynomial methods since the plane waves or bessel functions used in these methods are exact solutions of the helmholtz equation results indicate that this expectation does not necessarily translate into a clear benefit and that the differences in performance accuracy and conditioning are more nuanced than generally assumed the high order polynomial method can in fact deliver comparable and in some cases superior performance compared to the wave based dgm in addition to benchmarking the intrinsic computational performance of these methods a number of practical issues associated with realistic applications are also discussed

algorithmic construction of efficient fractional factorial designs with large run sizes fractional factorial ff designs are widely used in practice and typically are chosen according to the minimum aberration criterion a sequential algorithm is developed for constructing efficient ff designs a construction procedure is proposed that allows a design to be constructed only from its minimum aberration projection in the sequential buildup process to efficiently identify nonisomorphic designs designs are categorized according to moment projection pattern a fast isomorphism checking procedure is developed by matching the factors using their delete one factor projections this algorithm is used to completely enumerate all 128 run designs of resolution 4 all 256 run designs of resolution 4 up to 17 factors all 512 run designs of resolution 5 all 1024 run designs of resolution 6 and all 2048 and 4 096 run designs of resolution 7 a method is proposed for constructing minimum aberration ma designs using only a partial catalog of some good designs three approaches to constructing good de

regular fractional factorial designs with minimum aberration and maximum estimation capacity using the approach of finite projective geometry we make a systematic study of estimation capacity a criterion of model robustness under the absence of interactions involving three or more factors some general results providing designs with maximum estimation capacity are obtained in particular for two level factorials it is seen that constructing a design with maximum estimation capacity calls for choosing points from a finite projective geometry such that the number of lines is maximized and the distribution of these lines among the chosen points is as uniform as possible we also explore the connection with minimum aberration designs under which the sizes of the alias sets of two factor interactions which are not aliased with main effects are the most uniform possible

green function discretization for numerical solution of the helmholtz equation abstract a discretization method based on free space green function interpolation with a minimuml2norm property has been developed for numerical solution of the helmholtz equation it has been shown to produce accurate solutions with less than three points per wave which implies a manyfold improvement in solution efficiency over previous methods the accuracy of the method does not break down formally until the grid resolution approaches two points per wave

discontinuous galerkin methods with plane waves for the displacement based acoustic equation several special finite element methods have been proposed to solve helmholtz problems in the mid frequency regime such as the partition of unity method the ultra weak variational formulation and the discontinuous enrichment method the first main purpose of this paper is to present a discontinuous galerkin method with plane waves which is a variant of the discontinuous enrichment method to solve the displacement based acoustic equation the use of the displacement variable is often necessary in the context of fluid structure interactions a well known issue with this model is the presence of spurious vortical modes when one uses standard finite elements such as lagrange elements this problem also known as the locking phenomenon is observed with several other vector based equations such as incompressible elasticity and electromagnetism so this paper also aims at assessing if the special finite element methods suffer from the locking phenomenon in the context of the displacement acoustic equation the discontinuous galerkin method presented in this paper is shown to be very accurate and stable i e no spurious modes are observed the optimal choice of the various parameters are discussed with regards to numerical accuracy and conditioning some interesting properties of the mixed displacement pressure formulation are also presented furthermore the use of the partition of unity method is also presented but it is found that spurious vortical modes may appear with this method

minimum aberration fractional factorial designs with large n xu has cataloged 165 minimum aberration ma regular fractional factorial ff designs with 2 levels and large run sizes n 128 m 8 64 factors n 256 m 9 28 109 119 n 512 m 10 25 238 246 n 1024 m 11 24 488 501 n 2048 m 12 23 and n 4096 m 13 24 such an extensive catalog was produced because of an improved algorithm we extend the catalog by 36 ma 2 level regular ff designs n 256 m 29 36 100 108 n 512 m 26 29 n 1024 m 25 28 n 2048 m 24 32 and n 4096 m 25 26 although such enumeration problems are notoriously difficult with increased n and or m we brought the newly solved problems within computational reach by changing the vital isomorphism check component of xu s algorithm here we present a new compact graph for solving regular design isomorphism problems and use the nauty program to solve the corresponding graph isomorphism problems supplemental appendices are available online

a general rotation method for orthogonal latin hypercubes summaryorthogonal latin hypercubes provide a class of useful designs for computer experiments among the available methods for constructing such designs the method of rotation is particularly prominent due to its theoretical appeal as well as its space filling properties this paper presents a general method of rotation for constructing orthogonal latin hypercubes making the rotation idea applicable to many more situations than the original method allows in addition to general theoretical results many new orthogonal latin hypercubes are obtained and tabulated

an integral formulation for wave propagation on weakly non uniform potential flows abstract an integral formulation for acoustic radiation in moving flows is presented it is based on a potential formulation for acoustic radiation on weakly non uniform subsonic mean flows this work is motivated by the absence of suitable kernels for wave propagation on non uniform flow the integral solution is formulated using a green s function obtained by combining the taylor and lorentz transformations although most conventional approaches based on either transform solve the helmholtz problem in a transformed domain the current green s function and associated integral equation are derived in the physical space a dimensional error analysis is developed to identify the limitations of the current formulation numerical applications are performed to assess the accuracy of the integral solution it is tested as a means of extrapolating a numerical solution available on the outer boundary of a domain to the far field and as a means of solving scattering problems by rigid surfaces in non uniform flows the results show that the error associated with the physical model deteriorates with increasing frequency and mean flow mach number however the error is generated only in the domain where mean flow non uniformities are significant and is constant in regions where the flow is uniform

the ultra weak variational formulation for elastic wave problems the ultra weak variational formulation has been used effectively to solve time harmonic acoustic and electromagnetic wave propagation in inhomogeneous media we develop the ultra weak variational formulation for elastic wave propagation in two space dimensions in order to improve the accuracy and stability of the method we find it necessary to approximate the s and p wave components of the solution in a balanced way some preliminary analysis is provided and numerical evidence is presented for the efficiency of the scheme in comparison to piecewise linear finite elements

discontinuous galerkin methods with plane waves for time harmonic problems a general framework for discontinuous galerkin methods in the frequency domain with numerical flux is presented the main feature of the method is the use of plane waves instead of polynomials to approximate the solution in each element the method is formulated for a general system of linear hyperbolic equations and is applied to problems of aeroacoustic propagation by solving the two dimensional linearized euler equations it is found that the method requires only a small number of elements per wavelength to obtain accurate solutions and that it is more efficient than high order drp schemes in addition the conditioning of the method is found to be high but not critical in practice it is shown that the ultra weak variational formulation is in fact a subset of the present discontinuous galerkin method a special extension of the method is devised in order to deal with singular solutions generated by point sources like monopoles or dipoles aeroacoustic problems with non uniform flows are also considered and results are presented for the sound radiated from a two dimensional jet

orthogonal arrays robust to nonnegligible two factor interactions summary regular fractional factorial designs with clear two factor interactions provide a useful class of designs that are robust to nonnegligible two factor interactions in this paper the concept of clear two factor interactions is generalised to orthogonal arrays the new concept leads to a much wider class of designs robust to nonnegligible two factor interactions we study the existence and construction of such designs the designs we construct have a structure that render themselves particularly attractive in the robust parameter design setting we also discuss an interesting connection between designs with clear two factor interactions and mixed orthogonal arrays

a symmetric trefftz dg formulation based on a local boundary element method for the solution of the helmholtz equation a general symmetric trefftz discontinuous galerkin method is built for solving the helmholtz equation with piecewise constant coefficients the construction of the corresponding local solutions to the helmholtz equation is based on a boundary element method a series of numerical experiments displays an excellent stability of the method relatively to the penalty parameters and more importantly its outstanding ability to reduce the instabilities known as the pollution effect in the literature on numerical simulations of long range wave propagation

some results on 2n m designs of resolution iv with weak minimum aberration it is known that all resolution iv regular 2 n m designs of run size n 2 n m where 5n 16 n n 2 must be projections of the maximal even design with n 2 factors and therefore are even designs this paper derives a general and explicit relationship between the wordlength pattern of any even 2 n m design and that of its complement in the maximal even design using these identities we identify some weak minimum aberration 2 n m designs of resolution iv and the structures of their complementary designs based on these results several families of minimum aberration 2 n m designs of resolution iv are constructed

finite element solution of the helmholtz equation with high wave number part ii the h p version of the fem in this paper which is part ii in a series of two the investigation of the galerkin finite element solution to the helmholtz equation is continued while part i contained results on the h version with piecewise linear approximation the present part deals with approximation spaces of order p ge 1 as in part i the results are presented on a one dimensional model problem with dirichlet robin boundary conditions in particular there are proven stability estimates both with respect to data of higher regularity and data that is bounded in lower norms the estimates are shown both for the continuous and the discrete spaces under consideration further there is proven a result on the phase difference between the exact and the galerkin finite element solutions for arbitrary p that had been previously conjectured from numerical experiments these results and further preparatory statements are then employed to show error estimates for the galerkin finite element method fem it becomes evident that the error estimate for higher approximation can with certain assumptions on the data be written in the same form as the piecewise linear case namely as the sum of the error of best approximation plus a pollution term that is of the order of the phase difference the paper is concluded with a numerical evaluation

discontinuous galerkin method based on non polynomial approximation spaces in this paper we develop discontinuous galerkin dg methods based on non polynomial approximation spaces for numerically solving time dependent hyperbolic and parabolic and steady state hyperbolic and elliptic partial differential equations pdes the algorithm is based on approximation spaces consisting of non polynomial elementary functions such as exponential functions trigonometric functions etc with the objective of obtaining better approximations for specific types of pdes and initial and boundary conditions it is shown that l2 stability and error estimates can be obtained when the approximation space is suitably selected it is also shown with numerical examples that a careful selection of the approximation space to fit individual pde and initial and boundary conditions often provides more accurate results than the dg methods based on the polynomial approximation spaces of the same order of accuracy

approximation in the mean by solutions of elliptic equations on demontre un resultat analogue au theoreme d approximation de vituskin pour l approximation en moyenne par des solutions de certaines equations elliptiques

some theory for constructing minimum aberration fractional factorial designs summary minimum aberration is the most established criterion for selecting a regular fractional factorial design of maximum resolution minimum aberration designs for n runs and n 2 m n factors have previously been constructed using the novel idea of complementary designs in this paper an alternative method of construction is developed by relating the wordlength pattern of designs to the so called confounding between experimental runs this allows minimum aberration designs to be constructed for n runs and 5n 16 m n 2 factors as well as for n 2 m n

a least squares method for the helmholtz equation abstract we investigate the use of least squares methods to approximate the helmholtz equation the basis used in the discrete method consists of solutions of the helmholtz equation either consisting of plane waves or bessel functions on each element of a finite element grid unlike previous methods of this type we do not use polynomial based finite elements the use of small elements and relatively few basis functions per element allows us to prove convergence theorems for the method and to some extent control the conditioning of the resulting linear sy stem numerical results show the efficiency of the new method and suggest that it may be possible to obtain accurate results with a coarser grid than is usual for standard finite element methods

special short wave elements for flow acoustics a numerical formulation based on the partition of unity method pum is proposed for modelling the propagation of short acoustic waves on irrotational mean flows the method seeks to reduce the pollution error which exists in conventional fe schemes at high frequencies by using a local basis which is enriched by plane wave solutions of the convected helmholtz equation initially the method is demonstrated with reference to a one dimensional model consisting of a variable area converging diverging duct with mean flow next a simple two dimensional model of a straight duct with uniform flow is considered in both cases the accuracy and the conditioning of the numerical solution is investigated for ranges of frequency and mach number characteristic of aero engine bypass ducts

a new category of hermitian upwind schemes for computational acoustics a new high order upwind scheme is developed to solve linear acoustics in order to produce a compact scheme with a high level of precision a hermite interpolation is used an algorithm in three stages is built in order to optimize the procedure of discretization in the reconstruction stage a local spatial interpolator based on an upwind stencil is defined in the decomposition stage the calculation of a wave model adapted to the decomposition of the time derivatives in simple wave contributions is derived then the local spatial interpolator is used in order to calculate the wave strengths for each wave in the evolution stage we update by means of a procedure similar to that of cauchy kovaleskaya the discrete variable and its derivatives by using the information brought by each simple wave specific boundary conditions are finally established in this way a third order upwind hermitian scheme may be constructed d p3 scheme a one dimensional analysis in fourier series shows the principal properties of such a scheme the effectiveness and the exactitude of the new compact scheme are shown by their applications to several two dimensional problems and by comparisons to competing finite difference method for wave propagation

the partition of unity finite element method for short wave acoustic propagation on non uniform potential flows a novel numerical method is proposed for modelling time harmonic acoustic propagation of short wavelength disturbances on non uniform potential flows the method is based on the partition of unity finite element method in which a local basis of discrete plane waves is used to enrich the conventional finite element approximation space the basis functions are local solutions of the governing equations they are able to represent accurately the highly oscillatory behaviour of the solution within each element while taking into account the convective effect of the flow and the spatial variation in local sound speed when the flow is non uniform many wavelengths can be included within a single element leading to ultra sparse meshes results presented in this article will demonstrate that accurate solutions can be obtained in this way for a greatly reduced number of degrees of freedom when compared to conventional element or grid based schemes numerical results for lined uniform two dimensional ducts and for non uniform axisymmetric ducts are presented to indicate the accuracy and performance which can be achieved numerical studies indicate that the pollution effect associated with cumulative dispersion error in conventional finite element schemes is largely eliminated

implementation of an interior point source in the ultra weak variational formulation through source extraction the ultra weak variational formulation uwvf is a powerful numerical method for the approximation of acoustic elastic and electromagnetic waves in the time harmonic regime the use of trefftz type basis functions incorporates the known wave like behaviour of the solution in the discrete space allowing large reductions in the required number of degrees of freedom for a given accuracy when compared to standard finite element methods however the uwvf is not well disposed to the accurate approximation of singular sources in the interior of the computational domain we propose an adjustment to the uwvf for seismic imaging applications which we call the source extraction uwvf differing fields are solved for in subdomains around the source and matched on the inter domain boundaries numerical results are presented for a domain of constant wavenumber and for a domain of varying sound speed in a model used for seismic imaging

robust designs through partially clear two factor interactions summary orthogonal arrays with clear two factor interactions provide a class of designs that are robust to nonnegligible effects if certain prior knowledge is available then robust designs allow additional factors to be studied this is done through partially clear two factor interactions we study the existence and construction of such robust designs and present an upper bound on the maximum number of clear two factor interactions

higher order extensions of a discontinuous galerkin method for mid frequency helmholtz problems recently a discontinuous galerkin method with plane wave basis functions and lagrange multiplier degrees of freedom was proposed for the efficient solution of helmholtz problems in the mid frequency regime in this paper this method is extended to higher order elements performance results obtained for various two dimensional problems highlight the advantages of these elements over classical higher order galerkin elements such as q2 and q4 for the discretization of interior and exterior helmholtz problems copyright 2004 john wiley sons ltd

a phase based hybridizable discontinuous galerkin method for the numerical solution of the helmholtz equation we introduce a new hybridizable discontinuous galerkin hdg method for the numerical solution of the helmholtz equation over a wide range of wave frequencies our approach combines the hdg methodology with geometrical optics in a fashion that allows us to take advantage of the strengths of these two methodologies the phase based hdg method is devised as follows first we enrich the local approximation spaces with precomputed phases which are solutions of the eikonal equation in geometrical optics second we propose a novel scheme that combines the hdg method with ray tracing to compute multivalued solution of the eikonal equation third we utilize the proper orthogonal decomposition to remove redundant modes and obtain locally orthogonal basis functions which are then used to construct the global approximation spaces of the phase based hdg method and fourth we propose an appropriate choice of the stabilization parameter to guarantee stability and accuracy for the proposed method numerical experiments presented show that optimal orders of convergence are achieved that the number of degrees of freedom to achieve a given accuracy is independent of the wave number and that the number of unknowns required to achieve a given accuracy with the proposed method is orders of magnitude smaller than that with the standard finite element method

2n l designs with weak minimum aberration since not all 2 n l fractional factorial designs with maximum resolution are equally good fries and hunter introduced the minimum aberration criterion for selecting good 2 n l fractional factorial designs with the same resolution we modify the concept of minimum aberration and define weak minimum aberration and show the usefulness of this new design concept using some techniques from finite geometry we construct 2 n l fractional factorial designs of resolution iii with weak minimum aberration further several families of 2 n l fractional factorial designs of resolution iii and iv with minimum aberration are obtained

an improved partition of unity finite element model for diffraction problems the partition of unity finite element method pufem is explored and improved to deal with practical diffraction problems efficiently the use of plane waves as an external function space allows an efficient implementation of an approximate exterior non reflective boundary condition improving the original proposed by higdon for general diffraction problems a virtually analytical integration procedure is introduced for multi dimensional high frequency problems which exhibits a dramatic decrease in the number of operations for a given error compared with standard integration methods suitable conjugate gradient type solvers for the whole range of wavenumbers are used including such cases in which pufem can produce nearly singular matrices caused by round off limits copyright 2001 john wiley sons ltd

construction properties and analysis of group orthogonal supersaturated designs in this article we propose a new method for constructing supersaturated designs that is based on the kronecker product of two carefully chosen matrices the construction method leads to a partitio

application of an ultra weak variational formulation of elliptic pdes to the two dimensional helmholtz problem a new technique to solve elliptic linear pdes called ultra weak variational formulation uwvf in this paper is introduced in b despres c r acad sci paris 318 1994 pp 939 944 this paper is devoted to an evaluation of the potentialities of this technique it is applied to a model wave problem the two dimensional helmholtz problem the new method is presented in three parts following the same style of presentation as the classical one of the finite elements method even though they are definitely conceptually different methods the first part is committed to the variational formulation and to the continuous problem the second part defines the discretization process using a galerkin procedure the third part actually studies the efficiency of the technique from the order of convergence point of view this is achieved using theoretical proofs and a series of numerical experiments in particular it is proven and shown the order of convergence is lower bounded by a linear function of the number of degrees of freedom an application to scattering problems is presented in a fourth part

a discontinuous galerkin method with plane waves for sound absorbing materials poro elastic materials are commonly used for passive control of noise and vibration and are key to reducing noise emissions in many engineering applications including the aerospace automotive and energy industries more efficient computational models are required to further optimise the use of such materials in this paper we present a discontinuous galerkin method dgm with plane waves for poro elastic materials using the biot theory solved in the frequency domain this approach offers significant gains in computational efficiency and is simple to implement costly numerical quadratures of highly oscillatory integrals are not needed it is shown that the biot equations can be easily cast as a set of conservation equations suitable for the formulation of the wave based dgm a key contribution is a general formulation of boundary conditions as well as coupling conditions between different propagation media this is particularly important when modelling porous materials as they are generally coupled with other media such as the surround fluid or an elastic structure the validation of the method is described first for a simple wave propagating through a porous material and then for the scattering of an acoustic wave by a porous cylinder the accuracy conditioning and computational cost of the method are assessed and comparison with the standard finite element method is included it is found that the benefits of the wave based dgm are fully realised for the biot equations and that the numerical model is able to accurately capture both the oscillations and the rapid attenuation of the waves in the porous material

a comparison of wave based discontinuous galerkin ultra weak and least square methods for wave problems several numerical methods using non polynomial interpolation have been proposed for wave propagation problems at high frequencies the common feature of these methods is that in each element the solution is approximated by a set of local solutions they can provide very accurate solutions with a much smaller number of degrees of freedom compared to polynomial interpolation there are however significant differences in the way the matching conditions enforcing the continuity of the solution between elements can be formulated the similarities and discrepancies between several non polynomial numerical methods are discussed in the context of the helmholtz equation the present comparison is concerned with the ultra weak variational formulation uwvf the least squares method lsm and the discontinuous galerkin method with numerical flux dgm an analysis in terms of trefftz methods provides an interesting insight into the properties of these methods second the uwvf and the lsm are reformulated in a similar fashion to that of the dgm this offers a unified framework to understand the properties of several non polynomial methods numerical results are also presented to put in perspective the relative accuracy of the methods the numerical accuracies of the methods are compared with the interpolation errors of the wave bases

a comparison of two trefftz type methods the ultraweak variational formulation and the least squares method for solving shortwave 2 d helmholtz problems trefftz methods for the numerical solution of partial differential equations pdes on a given domain involve trial functions which are defined in subdomains are generally discontinuous and are solutions of the governing pde or its adjoint within each subdomain the boundary conditions and matching conditions between subdomains must be enforced separately an interesting novel result presented in this paper is that the least squares method lsm and the ultraweak variational formulation two methods already established for solving the helmholtz equation can be derived in the framework of the trefftz type methods in the first case the boundary conditions and interelement continuity are enforced by means of a least squares procedure in the second a galerkin type weighted residual method is used another goal of the work is to assess the relative efficiency of each method for solving shortwave problems in acoustics and to study the stability of each method the numerical performance of each scheme is assessed with reference to two 2 d test problems acoustic propagation in an uniform soft walled duct and propagation in an l shaped domain the latter involving singular behaviour at a sharp corner

minimum aberration and model robustness for two level fractional factorial designs summary the performance of minimum aberration two level fractional factorial designs is studied under two criteria of model robustness simple sufficient conditions for a design to dominate another design with respect to each of these two criteria are derived it is also shown that a minimum aberration design of resolution ilil or higher maximizes the number of two factor interactions which are not aliases of main effects and subject to that condition minimizes the sum of squares of the sizes of alias sets of two factor interactions this roughly says that minimum aberration designs tend to make the sizes of the alias sets very uniform it follows that minimum aberration is a good surrogate for the two criteria of model robustness that are studied here examples are given to show that minimum aberration designs are indeed highly efficient

orthogonal and nearly orthogonal designs for computer experiments we introduce a method for constructing a rich class of designs that are suitable for use in computer experiments the designs include latin hypercube designs and two level fractional factorial designs as special cases and fill the vast vacuum between these two familiar classes of designs the basic construction method is simple building a series of larger designs based on a given small design if the base design is orthogonal the resulting designs are orthogonal likewise if the base design is nearly orthogonal the resulting designs are nearly orthogonal we present two generalizations of our basic construction method the first generalization improves the projection properties of the basic method the second generalization gives rise to designs that have smaller correlations sample constructions are presented and properties of these designs are discussed

green s function discretization scheme for sound propagation in nonuniform flows a new frequency domain discretization technique that permits the numerical solution of the equation governing the propagation of small disturbances within nonuniform potential flows is described the method can be applied successfully to the numerical solution of aeroacoustic problems because a good accuracy is preserved up to 3 4 points per period for three dimensional unstructured meshes the discretization scheme is based on a local interpolation formula that is strictly joined to the physics of wave propagation because it is constructed with the superimposition of elementary sources that are local solutions of the local convective wave equation the method presents aspects in common with both finite difference and finite element methods with the peculiarity that the local interpolation formula can be interpreted as a specific shape function introduced to take advantage of the physics of the problem the method is applied to the potential equation linearized around an arbitrary aerodynamic mean flow and several comparisons with theoretical results as well as several convergence tests are conducted to show that a good accuracy is preserved up to 3 4 points per period also for irregular meshes with both random and systematic distortions numerical calculations are presented for three dimensional problems with both uniform and nonuniform flow and comparisons are made with theoretical and other available numerical results with the appropriate generalizations the method can be regarded as a new efficient general approach for the discretization of generic partial differential equations

the discontinuous enrichment method for multiscale analysis computation naturally separates scales of a problem according to the mesh size a variety of improved numerical methods are described by multiscale considerations differing in the treatment of the unresolved fine scales the discontinuous enrichment method provides a unique multiscale approach to computation by employing fine scales that contain solutions of the homogeneous partial differential equation in a discontinuous framework the method thus combines relative ease of implementation with improved numerical performance these properties are demonstrated for both multiscale wave and transport problems pointing to the potential of considerable savings in computational resources

a discontinuous galerkin method with lagrange multipliers for the solution of helmholtz problems in the mid frequency regime we present a discontinuous galerkin method dgm for the solution of the helmholtz equation in the mid frequency regime our approach is based on the discontinuous enrichment method in which the standard polynomial field is enriched within each finite element by a non conforming field that contains free space solutions of the homogeneous partial differential equation to be solved hence for the helmholtz equation the enrichment field is chosen here as the superposition of plane waves we enforce a weak continuity of these plane waves across the element interfaces by suitable lagrange multipliers preliminary results obtained for two dimensional model problems discretized by uniform meshes reveal that the proposed dgm enables the development of elements that are far more competitive than both the standard linear and the standard quadratic galerkin elements for the discretization of helmholtz problems

a numerical integration scheme for special finite elements for the helmholtz equation the theory for integrating the element matrices for rectangular triangular and quadrilateral finite elements for the solution of the helmholtz equation for very short waves is presented a numerical integration scheme is developed samples of maple and fortran code for the evaluation of integration abscissae and weights are made available the results are compared with those obtained using large numbers of gauss legendre integration points for a range of testing wave problems the results demonstrate that the method gives correct results which gives confidence in the procedures and show that large savings in computation time can be achieved

blocked regular fractional factorial designs with minimum aberration this paper considers the construction of minimum aberration ma blocked factorial designs based on coding theory the concept of minimum moment aberration due to xu 2003 for unblocked designs is extended to blocked designs the coding theory approach studies designs in a row wise fashion and therefore links blocked designs with nonregular and supersaturated designs a lower bound on blocked wordlength pattern is established it is shown that a blocked design has ma if it originates from an unblocked ma design and achieves the lower bound it is also shown that a regular design can be partitioned into maximal blocks if and only if it contains a row without zeros sufficient conditions are given for constructing ma blocked designs from unblocked ma designs the theory is then applied to construct ma blocked designs for all 32 runs 64 runs up to 32 factors and all 81 runs with respect to four combined wordlength patterns

computational aspects of the ultra weak variational formulation the ultra weak variational formulation uwvf approach has been proposed as an effective method for solving helmholtz problems with high wave numbers however for coarse meshes the method can suffer from instability in this paper we consider computational aspects of the ultra weak variational formulation for the inhomogeneous helmholtz problem we introduce a method to improve the uwvf scheme and we compare iterative solvers for the resulting linear system computations for the acoustic transmission problem in 2d show that the new approach enables helmholtz problems to be solved on a relatively coarse mesh for a wide range of wave numbers

the perfectly matched layer for the ultra weak variational formulation of the 3d helmholtz equation we investigate the feasibility of using the perfectly matched layer pml as an absorbing boundary condition for the ultra weak variational formulation uwvf of the 3d helmholtz equation the pml is derived using complex stretching of the spatial variables this leads to a modified helmholtz equation for which the uwvf can be derived in the standard discrete uwvf the approximating subspace is constructed from local solutions of the helmholtz equation in previous studies plane wave basis functions have been advocated because they simplify the building of the uwvf matrices for the pml domain we propose a special set of plane wave basis functions which allow fast computations and efficiently reduce spurious numerical reflections the method is validated by numerical experiments in comparison to a low order absorbing boundary condition the pml shows superior performance copyright 2004 john wiley sons ltd

c m approximation by solutions of elliptic equations and calderon zygmund operators soit xcr n compact et soit f une fonction deux fois continument differentiable dans un voisinage de x dont le laplacien s annule en chaque point de x alors il y a une suite f j telle que chaque f j est une fonction harmonique sur un voisinage dependant de j de x et f j f uniformement sur x pour o 2

plane wave discontinuous galerkin methods for the 2d helmholtz equation analysis of the p version plane wave discontinuous galerkin pwdg methods are a class of trefftz type methods for the spatial discretization of boundary value problems for the helmholtz operator delta omega 2 omega 0 they include the so called ultra weak variational formulation from o cessenat and b despres siam j numer anal 35 1998 pp 255 299 this paper is concerned with the a priori convergence analysis of pwdg in the case of p refinement that is the study of the asymptotic behavior of relevant error norms as the number of plane wave directions in the local trial spaces is increased for convex domains in two space dimensions we derive convergence rates employing mesh skeleton based norms duality techniques from p monk and d wang comput methods appl mech engrg 175 1999 pp 121 136 and plane wave approximation theory

characterization of minimum aberration 2 sp n k designs in terms of their complementary designs a general result is obtained that relates the word length pattern of a 2 n k design to that of its complementary design by applying this result and using group isomorphism we are able to characterize minimum aberration 2 n k designs in terms of properties of their complementary designs the approach is quite powerful for small values of 2 n k n 1 in particular we obtain minimum aberration 2 n k designs with 2 n k n 1 1 to 11 for any n and k

a complementary design theory for doubling a complementary design theory for doubling by hongquan xu 1 and ching shui cheng 2 university of california los angeles and university of california berkeley august 11 2006 chen and cheng 2006a discussed the method of doubling for con structing two level fractional factorial designs they showed that for 9n 32 n 5n 16 all minimum aberration designs with n runs and n factors are projections of the maximal design with 5n 16 factors which is constructed by repeatedly doubling the 2 5 1 design de ned by i abcde this paper develops a general complementary design the ory for doubling for any design obtained by repeated doubling general identities are established to link the wordlength patterns of each pair of complementary projection designs a rule is developed for choosing minimum aberration projection designs from the maximal design with 5n 16 factors it is further shown that for 17n 64 n 5n 16 all minimum aberration designs with n runs and n factors are projections of the maximal design with n runs and 5n 16 factors ams 2000 subject classi cations primary 62k15 key words and phrases maximal design minimum aberration pless power moment identity wordlength pattern running title doubling and complementary designs supported in part by nsf grant dms 0505728 supported in part by nsf grant dms 0505556

the discontinuous enrichment method we propose a finite element based discretization method in which the standard polynomial field is enriched within each element by a nonconforming field that is added to it the enrichment contains free space solutions of the homogeneous differential equation that are not represented by the underlying polynomial field continuity of the enrichment across element interfaces is enforced weakly by lagrange multipliers in this manner we expect to attain high coarse mesh accuracy without significant degradation of conditioning assuring good performance of the computation at any mesh resolution examples of application to acoustics and advection diffusion are presented

a general method of constructing e s2 optimal supersaturated designs there has been much recent interest in supersaturated designs and their application in factor screening experiments supersaturated designs have mainly been constructed by using the e s 2 optimality criterion originally proposed by booth and cox in 1962 however until now e s 2 optimal designs have only been established with certainty for n experimental runs when the number of factors m is a multiple of n 1 and in adjacent cases where m q n 1 r r 2 q an integer a method of constructing e s 2 optimal designs is presented which allows a reasonably complete solution to be found for various numbers of runs n including n 8 12 16 20 24 32 40 48 64

an adjoint based method for liner impedance eduction validation and numerical investigation abstract an inverse method for liner acoustic impedance eduction in a duct is presented the determination of adjoint equations allows to make explicit the directional derivatives of the objective function needed by the search algorithm direct and adjoint system of equations are spatially discretized by a discontinuous galerkin scheme the propagation model and the eduction process are successfully validated on nasa benchmark data extracted from literature in a no flow configuration objective functions written in terms of acoustic pressure and acoustic velocity are considered successively special attention is given to the possibilities offered by the method such as multiparameter identification with the study of a non homogeneous liner convergence history of the process is also examined revealing interesting properties for the velocity based objective function compared to the pressure based objective function in further developments the method is planned to be used with acoustic velocity fields obtained by laser doppler velocimetry

some identities on q sp n m designs with application to minimum aberration designs chen and hedayat and tang and wu studied and characterized minimum aberration 2 n m designs in terms of their complemetary designs based on a new and more powerful approach we extend the study to identify minimum aberration q n m designs through their complementary designs by using macwilliams identities and krawtchouk polynomials in coding theory we obtain some general and explicit relationships between the wordlength pattern of a q n m design and that of its complementary design these identities provide a powerful tool for characterizing minimum aberration q n m designs the case of q 3 is studied in more details

improvements for the ultra weak variational formulation summary in this paper we investigate strategies to improve the accuracy and efficiency of the ultra weak variational formulation uwvf of the helmholtz equation the uwvf is a trefftz type nonpolynomial method using basis functions derived from solutions of the adjoint helmholtz equation we shall consider three choices of basis function propagating plane waves original choice bessel basis functions and evanescent wave basis functions traditionally two dimensional triangular elements are used to discretize the computational domain however the element shapes affect the conditioning of the uwvf hence we investigate the use of different element shapes aiming to lower the condition number and number of degrees of freedom our results include the first tests of a plane wave method on meshes of mixed element types in many modeling problems evanescent waves occur naturally and are challenging to model therefore we introduce evanescent wave basis functions for the first time in the uwvf to tackle rapidly decaying wave modes the advantages of an evanescent wave basis are verified by numerical simulations on domains including curved interfaces copyright 2013 john wiley sons ltd

analysis of high order finite elements for convected wave propagation in this paper we examine the performance of high order finite element methods fem for aeroacoustic propagation based on the convected helmholtz equation a methodology is presented to measure the dispersion and amplitude errors of the p fem including non interpolating shape functions such as bubble shape functions a series of simple test cases are also presented to support the results of the dispersion analysis the main conclusion is that the properties of p fem that make its strength for standard acoustics e g exponential p convergence low dispersion error remain present for flow acoustics as well however the flow has a noticeable effect on the accuracy of the numerical solution even when the change in wavelength due to the mean flow is accounted for and an approximation of the dispersion error is proposed to describe the influence of the mean flow also discussed is the so called aliasing effect which can reduce the accuracy of the solution in the case of downstream propagation this can be avoided by an appropriate choice of mesh resolution

the ultra weak variational formulation of thin clamped plate problems we develop a new numerical scheme for a fourth order elliptic partial differential equation based on kirchhoff s thin plate theory in particular we extend the ultra weak variational formulation uwvf to thin plate problems with clamped plate boundary conditions the uwvf uses a finite element mesh and non polynomial basis functions after deriving the new method we then prove l 2 norm convergence on the boundary finally we investigate numerically the feasibility of the uwvf for both homogeneous and inhomogeneous problems and show examples of p and h convergence