gravimagnetic anomaly formulas for polyhedra of spatially linear media formulas for the gravity potential field and field gradient tensor are derived for a polyhedral target body of a spatially linear density medium the formulas also define the magnetic potential and field in the case of a medium of spatially linear magnetization this work generalizes existing solutions for the gravity field of a polyhedral target of linearly varying density the formulas are analyzed for singularities and for numerical error growth error growth with increasing target distance is found to be higher than in the corresponding uniform polyhedral case examination of the error sources reveals that some error reduction is possible on this basis new algorithms of improved error control over existing algorithms are proposed for the linear medium computational results confirm the expected improvement

new scheme for computing the magnetic field resulting from a uniformly magnetized arbitrary polyhedron the magnetic field at any point outside a uniformly magnetized polyhedron of arbitrary shape is obtained by adding the fields resulting from the effective free magnetic poles on each of the polygonal surfaces of the polyhedron for each polygonal surface the components of the field at the point of observation are expressed in terms of new line integrals around the edges of the polygon and the solid angle subtended by the polygon at the point of observation the line integrals are standard elementary forms this new approach makes the numerical evaluation of the magnetic fields for such models much simpler and faster than previously published methods

potential fields and their partial derivatives produced by a 2d homogeneous polygonal source a summary with some revisions we summarized and revised the present forward modeling methods for calculating the gravity and magnetic field components and their partial derivatives of a 2d homogeneous source with a polygonal cross section the responses of interest include the gravity field components and their first and second order partial derivatives and the magnetic field components and their first order partial derivatives the revised formulas consist of several basic quantities that are common in all the formulations a singularity appears when the observation point coincides with a polygon vertex this singularity is removable for the gravity formulas but not for the others the compact forms of the revised formulas make them easy to implement we compare the gravity and magnetic field components and their partial derivatives produced by a 2d prism whose polygonal cross section approximates a cylinder with the corresponding analytical fields and partial derivatives of the cylinder the perfect fittings presented by both data sets confirm the reliability of the updated formulas

noisy collective behaviour in deterministic cellular automata we investigate cellular automata in four and five dimensions for which chate and manneville recently have found nontrivial collective behaviour more precisely though being fully deterministic the average magnetization seems to be periodic respectively quasiperiodic with superimposed noise whose amplitude decreases with system size we confirm this behaviour on very large systems and over very large times we analyse in detail the statistical properties of the noise systems on small lattices and or subject to additional external noise are metastable arguments by grinstein et al suggest that in the periodic case the infinite deterministic systems should be metastable too these arguments are generalized to quasiperiodic systems we find evidence that they do indeed apply but we find no direct evidence for metastability of large systems

collective behaviour in a cellular automaton and isotropy of the neighbourhood abstract a three dimensional deterministic cellular automata rule 33 which exhibits a nontrivial quasiperiodic behaviour is studied we show that the way in which the neighbours are selected causes the appearance or disappearance of that behaviour

a totalistic three dimensional cellular automaton with quasiperiodic behaviour abstract a three dimensional deterministic totalistic cellular automation with non trivial quasiperiodic behaviour is presented different properties as intrinsic noise and influence of external noise are investigated an approach to an explanation of the phenomenon is discussed

on the singularities of the gravity field of a homogeneous polyhedral body the study of the gravity field of arbitrary polyhedral bodies of homogeneous density has provoked a series of publications over the last decades some of the researchers represented an arbitrary three dimensional body in terms of contours obtained by the intersection of horizontal planes with the body

magnetic gradients produced by a 2d homogeneous polygonal source we use a set of analytic equations for calculating horizontal and vertical gradients of the magnetic anomaly produced by a 2d homogeneous magnetic source with a polygonal cross section the magnetization vector of the source is assumed to be constant and known and the anomalous field is assumed to be much smaller than the ambient geomagnetic field in magnitude the derivation begins from an expression of the magnetic potential then we evaluate clockwise integrations to establish the equations a comparison between this analytic approach and a numerical approach using the finite difference method to compute gradients indicates the reliability of the equations the proposed method is numerically efficient and does not present singularity on the edges of the polygon prism except at the vertices where the gradients approach to infinity

periodic behavior of cellular automata i propose an explanation of the observation of a globally synchronized behavior of deterministic cellular automata and coupled map lattices together with local fluctuations

closed form formula of magnetic gradient tensor for a homogeneous polyhedral magnetic target a tetrahedral grid example abstracta closed form formula is developed for the full magnetic gradient tensor of a polyhedral body with a homogeneous magnetization vector it is based on the direct derivative technique on the closed form of the magnetic field these analytical expressions are implemented into an easy to use c package which simultaneously calculates the magnetic potential the magnetic field and the full magnetic gradient tensor for magnetic targets modern unstructured tetrahedral grids are adopted to represent the polyhedral body so that our code can deal with arbitrarily complicated magnetic targets a prismatic body is tested to verify the accuracies of our closed form formula excellent agreements are obtained between our closed form solutions and solutions of a prismatic magnetic body with differences up to machine precision a pipeline model is used to demonstrate its capability to deal with complicated magnetic targets this c code is freely available to the magnetic exploration community

high precision fourier forward modeling of potential fields we analyzed the numerical forward methods in the fourier domain for potential fields existing fourier domain forward methods applied the standard fast fourier transform fft algorithm to inverse transform a conjugate symmetrical spectrum into a real field it had significant speed advantages over space domain forward methods but suffered from problems including aliasing imposed periodicity and edge effect usually grid expansion was needed to reduce these errors which was equivalent to the numerical evaluation of the oscillatory fourier integral using the trapezoidal rule with smaller steps we tested a high precision fourier domain forward method based on a combined use of shift sampling technique and gaussian quadrature theory the trapezoidal rule applied by the standard fft algorithm to evaluate the continuous fourier transform was modified by introducing a shift parameter by choosing optimum values of as gaussian quadrature nodes we developed a gauss fft method for fourier forward modeling of potential fields no grid expansion was needed the sources can be set near the boundary of the fields or even go beyond the boundary the gauss fft method converged to the space domain solution much faster than the standard fft method with grid expansion forward modeling results almost identical to space domain ones can be obtained in less time numerical examples of both simple and complex 2d and 3d source forward modeling revealed the reliability and adaptability of the method

comparison of gravimetric formulas for uniform polyhedra the gravitational anomaly of a polyhedral target of uniform density is amenable to analytical solution and has received wide attention we analyze a number of apparently dissimilar solutions for consistency and comment on their suitability for numeric computation consistency is determined by evaluating a method over a special contour for which all methods must agree numerical performance is assessed through operation counts and algorithm classification into vertex line or surface type each type having a known accuracy bound for a given floating point precision we verify performance predictions through model calculations in particular we demonstrate the effectiveness of revised vertex and line algorithms and of a new analytical surface method our scheme of analysis and classification has general application it provides criteria by which existing and novel solutions may be examined for consistency and numerical performance

arbitrary precision frequencies of a free rectangular thin plate abstract a variational method developed by the authors and named whole element method wem is used to find the arbitrary precision frequencies of a rectangular thin plate within the germain lagrange theory having its four borders free of constraint wem consists is proposing an adequate functional and a sequence representing the plate transversal displacementw x y such a sequence is made of a linear combination of functions belonging to a complete set inl2 the sequence and not each co ordinate function is required to satisfy the essential or geometric conditions the sequence generation is systematic and no analysis of the classical natural modes of the plate is needed in particular trigonometric functions which a priori belong to a complete set in the domain are used in the present analysis the solving equations involving very simple sums arise from the minimization of the functional wem is based on theorems which show the ultimate exactness of the eigenvalues and the uniform convergence of the essential functions of the problem to the authors knowledge this problem has no classical solution

gravimagnetic similarity in anomaly formulas for uniform polyhedra gravitational and magnetic anomalies of an arbitrary target body are linked through poisson s differential relation for a uniform polyhedral target poisson s relation reduces to an algebraic link between gravity and magnetic anomaly formulas the derivation is given in tensor form it identifies for each target facet edge a vector function in terms of which the gravitational and magnetic potential and field anomaly formulas are similarly expressed as appropriately weighted linear combinations this similarity unifies the theory of uniform polyhedral anomalies it benefits analysis and construction of software that naturally embraces all anomalies in a single code the analysis is exemplified by a discussion of singularities and by the adaptation of three gravity field algorithms to the remaining gravitational and magnetic cases while retaining the respective computational advantages of the former gravity field algorithms

conjugate complex variables method for the computation of gravity anomalies using conjugate complex variables a generalized method is presented to derive formulas to calculate first and higher order derivatives of the gravity potential due to selected mass models double integrals in the computation of gravity gradient anomalies are transformed into complex contour integrals analytical expressions for higher order derivatives of the gravitational potential in arbitrary directions due to two dimensional 2 d polygonal mass models are derived the method is extended to 2 d polygonal bodies whose density contrasts vary with depth and horizontal distance and can be generalized to deal with 2 d bodies of any shape the vertical gravity field and its first derivatives due to a homogeneous radially symmetric body are also computed using conjugate complex variables derivation of gravity and gravity gradient formulas generally is greatly simplified by the use of complex variables

3d vector gravity potential and line integrals for the gravity anomaly of a rectangular prism with 3d variable density contrast three dimensional rectangular prisms are building blocks for calculating gravity anomalies from irregular 3d mass bodies with spatially variable density contrasts a 3d vector gravity potential is defined for a 3d rectangular prism with density contrast varying in depth and horizontally the vertical component of the gravity anomaly equals the flux of the 3d vector gravity potential through the enclosed surface of the prism thus the 3d integral for the gravity anomaly is reduced to a 2d surface integral in turn a 2d vector gravity potential is defined the vertical component of the gravity anomaly equals the net circulation of the 2d vector gravity potential along the enclosed contour bounding the surfaces of the prism the 3d integral for the gravity anomaly is reduced to 1d line integrals further analytical or numerical solutions can then be obtained from the line integrals depending on the forms of the density contrast functions if an analytical solution cannot be obtained the line integral method is semianalytical requiring numerical quadratures to be carried out at the final stages singularity and discontinuity exist in the algorithm and the method of exclusive infinitesimal sphere or circle is effective to remove them then the vector potential line integral method can calculate the gravity anomaly resulting from a rectangular prism with density contrast varying simply in one direction and sophisticatedly in three directions the advantage of the method is that the constraint to the form of the density contrast is greatly reduced and the numerical calculation for the gravity anomaly is fast

elastic and piezoelectric fields due to polyhedral inclusions abstract we derive explicit closed form expressions describing elastic and piezoelectric deformations due to polyhedral inclusions in uniform half space and bi materials our analysis is based on the linear elasticity theory and green s function method the method involves evaluation of volume and surface integrals of harmonic and bi harmonic potentials in case of polyhedra such integrals are expressed through algebraic functions our results generalize numerous studies on this subject and they allow to obtain fully analytical solutions for a number of physical and engineering problems in the limiting case of an infinite space our relations have an essentially more compact form than relations obtained by other authors we present solutions to classical mindlin and cherruti problems we describe the elastic relaxation of a misfitting polygonal quantum dot in bi materials assuming isotropic and vertically isotropic properties it is explained how to analyze non hydrostatic and non uniform inclusions we also study piezoelectric fields induced by inclusions in materials with cubic and hexagonal lattices among other results we have found that a cubic inclusion in an isotropic material reproduces fields of quantum dots in gaas 0 0 1 and gaas 1 1 1 depending on the orientation of the cube this suggests that one can qualitatively model crystals with different lattices by choosing an appropriate inclusion shape

gravimetric analysis of uniform polyhedra analytical formulas for the gravity anomaly of a uniform polyhedral body are subject to numerical error that increases with distance from the target while the anomaly decreases this leads to a limited range of target distances in which the formulas are operational beyond which the calculations are dominated by rounding error we analyze the sources of error and propose a combination of numerical and analytical procedures that exhibit advantages over existing methods namely 1 errors that diminish with distance 2 enhanced operating range and 3 algorithmic simplicity the latter is achieved by avoiding the need to transform coordinates and the need to discriminate between projected observation points that lie inside on or outside a target facet boundary our error analysis is verified in computations based on a published code and on a code implementing our methods the former requires a numerical precision of one part in 1016 double precision in problems of geophysical interest whereas our c

new method for fast computation of gravity and magnetic anomalies from arbitrary polyhedra we show that at any point the gravity field from a solid body bounded by plane surfaces and having uniform density can be computed as a field from a fictitious distribution of surface mass density on the same body the surface mass density at every surface element is equal to the product of the volume density of the body and the scalar product of 1 the unit outward vector normal to that surface element and 2 the position vector of the surface element with respect to the point of observation accordingly the contribution to the gravity field from any plane surface of the body vanishes if the observation point lies in the plane of that surface as a result we can compute the gravity field everywhere including points inside on the surface on an edge or at a corner of the body where more than two surfaces meet this new result lets us compute the gravity field using exactly the same simple procedure as for the magnetic field of a uniformly magnetized object computed from an equivalent surface distr

vibration of non homogeneous rectangular membranes with arbitrary interfaces abstract the study deals with the generalized solution of the title problem the free vibration problem of a rectangular membrane with partial domains each of uniform density and arbitrary interface is tackled previous studies by other researchers include straight parallel to the borders and oblique interfaces and bent ones the solution is found by means of a direct variational method with a series composed with a complete set of functions two alternative sets are explored trigonometric and power series such series are uniformly convergent to the exact solution the approach is straightforward and very efficient from the computational viewpoint a determinant factorization method is employed to automatically eliminate eventual spurious frequencies the well known analogy between plates and membranes does not hold in this problem and a demonstration is included diverse illustrations are worked out as the cases of an oblique straight line interface and an open curve line which divides the membrane in two domains each of different density are first presented also a rectangular membrane with an interior closed domain is stated and the numerical example of a circular interior zone is included comparison between the two alternatives and with other authors results show excellent agreement in all cases the computational cost is very low