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Linear Equations (linear + equation)
Selected AbstractsFlapwise bending vibration of rotating platesINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 7 2002H. H. Yoo Abstract Linear equations of motion for the flapwise bending vibration analysis of rotating plates are derived in the present work. The equations of motion are transformed into dimensionless forms in which three dimensionless parameters are identified. The effects of the dimensionless parameters on the characteristics of the flapwise bending vibration of rotating plates are investigated. The accuracy of the present modelling method is verified through comparing its numerical results to those obtained by an existing method in the related literature. Eigenvalue loci crossing and eigenvalue loci veering phenomena are observed and discussed. The variations of mode shapes associated with the phenomena are also exhibited. Copyright © 2002 John Wiley & Sons, Ltd. [source] Uncertainties in interpretation of isotope signals for estimation of fine root longevity: theoretical considerationsGLOBAL CHANGE BIOLOGY, Issue 7 2003YIQI LUOArticle first published online: 25 JUN 200 Abstract This paper examines uncertainties in the interpretation of isotope signals when estimating fine root longevity, particularly in forests. The isotope signals are depleted ,13C values from elevated CO2 experiments and enriched ,14C values from bomb 14C in atmospheric CO2. For the CO2 experiments, I explored the effects of six root mortality patterns (on,off, proportional, constant, normal, left skew, and right skew distributions), five levels of nonstructural carbohydrate (NSC) reserves, and increased root growth on root ,13C values after CO2 fumigation. My analysis indicates that fitting a linear equation to ,13C data provides unbiased estimates of longevity only if root mortality follows an on,off model, without dilution of isotope signals by pretreatment NSC reserves, and under a steady state between growth and death. If root mortality follows the other patterns, the linear extrapolation considerably overestimates root longevity. In contrast, fitting an exponential equation to ,13C data underestimates longevity with all the mortality patterns except the proportional one. With either linear or exponential extrapolation, dilution of isotope signals by pretreatment NSC reserves could result in overestimation of root longevity by several-fold. Root longevity is underestimated if elevated CO2 stimulates fine root growth. For the bomb 14C approach, I examined the effects of four mortality patterns (on,off, proportional, constant, and normal distribution) on root ,14C values. For a given ,14C value, the proportional pattern usually provides a shorter estimate of root longevity than the other patterns. Overall, we have to improve our understanding of root growth and mortality patterns and to measure NSC reserves in order to reduce uncertainties in estimated fine root longevity from isotope data. [source] Efficient implicit finite element analysis of sheet forming processesINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 8 2003A. H. van den Boogaard Abstract The computation time for implicit finite element analyses tends to increase disproportionally with increasing problem size. This is due to the repeated solution of linear sets of equations, if direct solvers are used. By using iterative linear equation solvers the total analysis time can be reduced for large systems. For plate or shell element models, however, the condition of the matrix is so ill that iterative solvers do not reach the huge time-savings that are realized with solid elements. By introducing inertial effects into the implicit finite element code the condition number can be improved and iterative solvers perform much better. An additional advantage is that the inertial effects stabilize the Newton,Raphson iterations. This also applies to quasi-static processes, for which the inertial effects finally do not affect the results. The presented method can readily be implemented in existing implicit finite element codes. Industrial size deep drawing simulations are executed to investigate the performance of the recommended strategy. It is concluded that the computation time is decreased by a factor of 5 to 10. Copyright © 2003 John Wiley & Sons, Ltd. [source] Solutions of linear and semilinear distributed parameter equations with a fractional Brownian motionINTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING, Issue 2 2009T. E. Duncan Abstract In this paper, some linear and semilinear distributed parameter equations (equations in a Hilbert space) with a (cylindrical) fractional Brownian motion are considered. Solutions and sample path properties of these solutions are given for the stochastic distributed parameter equations. The fractional Brownian motions are indexed by the Hurst parameter H,,,(0, 1). For H,=,˝ the process is Brownian motion. Solutions of these linear and semilinear equations are given for each H,,,(0, 1) with the assumptions differing for the cases H,,,(0, ˝) and H,,,(˝, 1). For the linear equations, the solutions are mild solutions and limiting Gaussian measures are characterized. For the semilinear equations, the solutions are either mild or weak. The weak solutions are obtained by transforming the measure of the associated linear equation by a Radon,Nikodym derivative (likelihood function). An application to identification is given by obtaining a strongly consistent family of estimators for an unknown parameter in a linear equation with distributed noise or boundary noise. Copyright © 2008 John Wiley & Sons, Ltd. [source] Placental thickness in the first half of pregnancyJOURNAL OF CLINICAL ULTRASOUND, Issue 5 2004Theera Tongsong MD Abstract Purpose This study was conducted to establish normal values of placental thickness during the first half of pregnancy. Methods Normal pregnant women with singleton pregnancies between 8 and 20 weeks of gestation were recruited into the study. All the newborns were normal at birth. Placental thickness was measured perpendicularly through the thickest part of the placenta on transabdominal scans. The placental thickness data were analyzed for mean, standard deviation, 95% confidence interval, and 2.5th, 5th, 50th, 95th, and 97.5th percentile for each week of gestational age. The best-fit mathematical model was derived by regression analysis. Results The total number of measurements was 333 and the number of measurements for each week of gestational age ranged from 9 to 37. Regression analysis yielded the following linear equation of the relationship: placental thickness (in mm) = gestational age (in weeks) × 1.4,5.6 (r = 0.82). Conclusion We have established a nomogram for placental thickness. This resource may be a useful aid in the early detection of placental abnormalities, such as hydropic placenta secondary to hemoglobin Bart's disease. © 2004 Wiley Periodicals, Inc. J Clin Ultrasound 32:231,234, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/jcu.20023 [source] Accurate prediction of thermodynamic properties of alkyl peroxides by combining density functional theory calculation with least-square calibrationJOURNAL OF COMPUTATIONAL CHEMISTRY, Issue 7 2009Cun-Xi Liu Abstract Owing to the significance in kinetic modeling of the oxidation and combustion mechanisms of hydrocarbons, a fast and relatively accurate method was developed for the prediction of ,fH of alkyl peroxides. By this method, a raw ,fH value was calculated from the optimized geometry and vibration frequencies at B3LYP/6-31G(d,p) level and then an accurate ,fH value was obtained by a least-square procedure. The least-square procedure is a six-parameter linear equation and is validated by a leave-one out technique, giving a cross-validation squared correlation coefficient q2 of 0.97 and a squared correlation coefficient of 0.98 for the final model. Calculated results demonstrated that the least-square calibration leads to a remarkable reduction of error and to the accurate ,fH values within the chemical accuracy of 8 kJ mol,1 except (CH3)2CHCH2CH2CH2OOH which has an error of 8.69 kJ mol,1. Comparison of the results by CBS-Q, CBS-QB3, G2, and G3 revealed that B3LYP/6-31G(d,p) in combination with a least-square calibration is reliable in the accurate prediction of the standard enthalpies of formation for alkyl peroxides. Standard entropies at 298 K and heat capacities in the temperature range of 300,1500 K for alkyl peroxides were also calculated using the rigid rotor-harmonic oscillator approximation. © 2008 Wiley Periodicals, Inc. J Comput Chem, 2009 [source] ELECTRICAL CONDUCTIVITY AND KINETIC PARAMETERS OF RICE STARCHJOURNAL OF FOOD PROCESS ENGINEERING, Issue 5 2007E. MORALES SANCHEZ ABSTRACT In the present work, rice starch was studied by differential scanning calorimetry (DSC) and electrical conductivity as a function of temperature. The gelatinization temperature was calculated using both methods. Three stages of electrical conductivity were found. When the logarithm of conductivity was plotted versus temperature, the data have been well adjusted to the linear equation, indicating that a model for conductivity may be created using an Arrhenius-type expression to obtain kinetic parameters such as onset, peakset, endset temperatures and gelatinization energy (Eg). The conductivity Eg was calculated and compared to the enthalpy H calculated from the DSC data. It was found that Eg and enthalpy are similar processes. We conclude that electrical conductivity is an alternative method for the study of starch,water mixtures. PRACTICAL APPLICATIONS This research provides a new methodology to obtain kinetic parameters such as temperature of the beginning, medium and ending of gelatinizaiton, which are important for determining the cooking range needed to obtain desired characteristics in a processed starch-containing food. [source] Debugging Decomposition Data,Comparative Taphonomic Studies and the Influence of Insects and Carcass Size on Decomposition RateJOURNAL OF FORENSIC SCIENCES, Issue 1 2010Tal Simmons Ph.D. Abstract:, Comparison of data from a variety of environments and ambient temperatures has previously been difficult as few studies used standardized measures of time/temperature and decomposition. In this paper, data from previous studies and recent experiments are compared using simple conversions. These conversions allow comparison across multiple environments and experiments for the first time. Plotting decomposition score against logADD allows the exponential progression of decomposition to be expressed as a simple linear equation. Data comparison from many environments and temperatures shows no difference in decomposition progression when measured using Accumulated Degree Days. The major effector of change in rate was insect presence, regardless of depositional environment, species, or season. Body size is significant when carcasses are accessed by insects; when insects are excluded, while bodies are indoors, submerged, or buried, then decomposition progresses at the same rate regardless of body size. [source] Effect of anesthetic structure on inhalation anesthesia: Implications for the mechanismJOURNAL OF PHARMACEUTICAL SCIENCES, Issue 6 2008Michael H. Abraham Abstract Many previous attempts (e.g., the Meyer,Overton hypothesis) to provide a single set of physical or chemical characteristics that accurately predict anesthetic potency have failed. A finding of a general predictive correlation would support the notion of a unitary theory of narcosis. Using the Abraham solvation parameter model, the minimum alveolar concentration, MAC, of 148 varied anesthetic agents can be fitted to a linear equation in log (1/MAC) with R2,=,0.985 and a standard deviation, SD,=,0.192 log units. Division of the 148 compounds into a training set and a test set shows that log (1/MAC) values can be predicted with no bias and with SD,=,0.20 log units. The two main factors that determine MAC values are compound size and compound hydrogen bond acidity, both of which increase anesthetic activity. Shape has little or no effect on anesthetic activity. Our observations support a unitary theory of narcosis by inhalation anesthetics. A two-stage mechanism for inhalation anesthesia accounts for the observed structural effects of anesthetics. In this mechanism, the first main step is transfer of the anesthetic to the site of action, and the second step is interaction of the anesthetic with a receptor(s). © 2007 Wiley-Liss, Inc. and the American Pharmacists Association J Pharm Sci 97:2373,2384, 2008 [source] Structure-stability correlations for imine formation in aqueous solutionJOURNAL OF PHYSICAL ORGANIC CHEMISTRY, Issue 10 2005C. Godoy-Alcántar Abstract Imine formation between 25 aldehydes and 13 amines in aqueous solution in the pH range 7,11 was studied by 1H NMR spectroscopy. A three-parameter linear equation correlating logarithms of imine formation constants with pKa and HOMO energies of amines and LUMO energies of aldehydes is proposed. In view of the widespread occurrence of imine-forming processes in both chemistry and biology, the data presented are of significance for physical organic chemistry and of particular interest for dynamic combinatorial chemistry. Copyright © 2005 John Wiley & Sons, Ltd. [source] Application of parameter differentiation for flow of a third grade fluid past an infinite porous plateNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 1 2010M. Sajid Abstract This article investigates the analytic solution for the flow of a third grade fluid past an infinite porous plate. The method of parameter differentiation is used to linearized the governing flow equation. The solution of the obtained linear equation is developed by differential transform method in combination with the method of superposition. The obtained results are compared with existing results in the literature and an excellent agreement is found. This shows that the parameter differentiation is a powerful technique for solving nonlinear problems. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010 [source] Optimal weighting design for distributed parameter systems estimationOPTIMAL CONTROL APPLICATIONS AND METHODS, Issue 1 2001Mostafa Ouarit Abstract This paper presents a method which aims at improving parameter estimation in dynamical systems. The general principle of the method is based on a modification of the least-squares objective function by means of a weighting operator, in view to improve the conditioning of the identification problem. First we recall a previous work using variational calculus in order to obtain the weighting operators through a linear equation. Then we propose a new approach which consists of determining the weights by formulating an optimization problem including positive semidefinite constraints (linear matrix inequalities, LMI). Copyright © 2001 John Wiley & Sons, Ltd. [source] Experimental Study on Particle Size Distribution and Concentration Using Transmission Fluctuation Spectrometry with the Autocorrelation TechniquePARTICLE & PARTICLE SYSTEMS CHARACTERIZATION, Issue 3 2005Xiaoai Guo Abstract Based on the statistical characteristics of the transmission fluctuations in the particle suspension, transmission fluctuation spectrometry with autocorrelation (TFS-AC) is described theoretically, with the assumptions of geometric ray propagation and completely absorbent particles in the suspension. The experiments presented here are realized in a focused Gaussian beam with the TFS-AC technique. The acquisition of transmission fluctuation signals is achieved by using a high-resolution digital oscilloscope. The transition function of TFS-AC is obtained by varying the autocorrelation time. With a modified iterative Chahine inversion algorithm, solving a linear equation retrieves information on the particle size distribution and particle concentration. Some experimental results on spherical and non-spherical particles are presented and discussed. The experiments cover a particle size range from 1,m to 1000,,m and a particle concentration of up to 12,%. [source] Bootstrap inference in a linear equation estimated by instrumental variablesTHE ECONOMETRICS JOURNAL, Issue 3 2008Russell Davidson Summary, We study several tests for the coefficient of the single right-hand-side endogenous variable in a linear equation estimated by instrumental variables. We show that writing all the test statistics,Student's t, Anderson,Rubin, the LM statistic of Kleibergen and Moreira (K), and likelihood ratio (LR),as functions of six random quantities leads to a number of interesting results about the properties of the tests under weak-instrument asymptotics. We then propose several new procedures for bootstrapping the three non-exact test statistics and also a new conditional bootstrap version of the LR test. These use more efficient estimates of the parameters of the reduced-form equation than existing procedures. When the best of these new procedures is used, both the K and conditional bootstrap LR tests have excellent performance under the null. However, power considerations suggest that the latter is probably the method of choice. [source] Adsorption, absorption, and biological degradation of ammonia in different biofilter organic mediaBIOTECHNOLOGY & BIOENGINEERING, Issue 3 2007Estela Pagans Abstract A tailor-made apparatus called ammoniometer, which is a batch mode respirometer applied to the study of ammonia biodegradation in biofilter media, has been used to evaluate adsorption, absorption, and biodegradation in five different organic materials (compost, coconut fibre, bark, pruning wastes, and peat) obtained from full-scale biofilters in operation in several waste treatment plants. The results showed that absorption could be represented by a Henry's law linear equation, with values of the Henry coefficient significantly higher (from 1,866 to 15,320) than that of pure water (1,498). Adsorption data were successfully fitted to Langmuir and Freundlich isotherms and maximum adsorption capacity varies from 1.06 to 1.81 mg NH3/g dry media. Ammonia biodegradation rates for each organic material were also calculated. Biodegradation rates varied from 0.67 to 7.82 mg NH3/kg media/d depending on the material tested. The data obtained showed important differences in the behaviour of the biofilter organic media, which has important implications in the design and modelling of these systems. Biotechnol. Bioeng. 2007;97: 515,525. © 2006 Wiley Periodicals, Inc. [source] A Novel Methodological Approach for the Analysis of Host,Ligand Interactions,CHEMPHYSCHEM, Issue 2 2007Daniela Strat Dr. Abstract Traditional analysis of drug-binding data relies upon the Scatchard formalism. These methods rely upon the fitting of a linear equation providing intercept and gradient data that relate to physical properties, such as the binding constant, cooperativity coefficients and number of binding sites. However, the existence of different binding modes with different binding constants makes the implementation of these models difficult. This article describes a novel approach to the binding model of host,ligand interactions by using a derived analytical function describing the observed signal. The benefit of this method is that physically significant parameters, that is, binding constants and number of binding sites, are automatically derived by the use of a minimisation routine. This methodology was utilised to analyse the interactions between a novel antitumour agent and DNA. An optical spectroscopy study confirms that the pentacyclic acridine derivative (DH208) binds to nucleic acids. Two binding modes can be identified: a stronger one that involves intercalation and a weaker one that involves oriented outer-sphere binding. In both cases the plane of the bound acridine ring is parallel to the nucleic acid bases, orthogonal to the phosphate backbone. Ultraviolet (UV) and circular dichroism (CD) data were fitted using the proposed model. The binding constants and the number of binding sites derived from the model remained consistent across the different techniques used. The different wavelengths at which the measurements were made maintained the coherence of the results. [source] Physiologically correct animation of the heartCOMPUTER ANIMATION AND VIRTUAL WORLDS (PREV: JNL OF VISUALISATION & COMPUTER ANIMATION), Issue 3-4 2008Kyoungju Park Abstract Physiologically correct animation of the heart should incorporate non-homogeneous and nonlinear motions of the heart. Therefore, we introduce a methodology that estimates deformations from volume images and utilizes them for animation. Since volume images are acquired at regular slicing intervals, they miss information between slices and recover deformation on the slices. Therefore, the estimated finite element models (FEMs) result in coarse meshes with chunk elements the sizes of which depend on the slice intervals. Thus, we introduce a method of generating a detailed model using implicit surfaces and transferring a deformation from a FEM to implicit surfaces. An implicit surface heart model is reconstructed using contour data points and then cross-parameterized to the heart FEM, the time-varying deformation of which has been estimated by tracking the insights of the heart wall. The implicit surface heart models are composed of four heart walls that are blended into one model. A correspondence map between the source and the target meshes is made using the template fitting method. Deformation coupling transfers the deformation of a coarse heart FEM model to a detailed implicit model by factorizing linear equations. We demonstrate the system and show the resulting deformation of an implicit heart model. Copyright © 2008 John Wiley & Sons, Ltd. [source] A fast triangle to triangle intersection test for collision detectionCOMPUTER ANIMATION AND VIRTUAL WORLDS (PREV: JNL OF VISUALISATION & COMPUTER ANIMATION), Issue 5 2006Oren Tropp Abstract The triangle-to-triangle intersection test is a basic component of all collision detection data structures and algorithms. This paper presents a fast method for testing whether two triangles embedded in three dimensions intersect. Our technique solves the basic sets of linear equations associated with the problem and exploits the strong relations between these sets to speed up their solution. Moreover, unlike previous techniques, with very little additional cost, the exact intersection coordinates can be determined. Finally, our technique uses general principles that can be applied to similar problems such as rectangle-to-rectangle intersection tests, and generally to problems where several equation sets are strongly related. We show that our algorithm saves about 20% of the mathematical operations used by the best previous triangle-to-triangle intersection algorithm. Our experiments also show that it runs 18.9% faster than the fastest previous algorithm on average for typical scenarios of collision detection (on Pentium 4). Copyright © 2006 John Wiley & Sons, Ltd. [source] Gradient Estimation in Volume Data using 4D Linear RegressionCOMPUTER GRAPHICS FORUM, Issue 3 2000László Neumann In this paper a new gradient estimation method is presented which is based on linear regression. Previous contextual shading techniques try to fit an approximate function to a set of surface points in the neighborhood of a given voxel. Therefore a system of linear equations has to be solved using the computationally expensive Gaussian elimination. In contrast, our method approximates the density function itself in a local neighborhood with a 3D regression hyperplane. This approach also leads to a system of linear equations but we will show that it can be solved with an efficient convolution. Our method provides at each voxel location the normal vector and the translation of the regression hyperplane which are considered as a gradient and a filtered density value respectively. Therefore this technique can be used for surface smoothing and gradient estimation at the same time. [source] Systematic lumped-parameter models for foundations based on polynomial-fraction approximationEARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS, Issue 7 2002Wen-Hwa Wu Abstract Based on the approximation by polynomial-fraction, a series of systematic lumped-parameter models are developed in this paper for efficiently representing the dynamic behaviour of unbounded soil. Concise formulation is first employed to represent the dynamic flexibility function of foundation with a ratio of two polynomials. By defining an appropriate quadratic error function, the optimal coefficients of the polynomials can be directly solved from a system of linear equations. Through performing partial-fraction expansion on this polynomial-fraction and designing two basic discrete-element models corresponding to the partial fractions, systematic lumped-parameter models can be conveniently established by connecting these basic units in series. Since the systematic lumped-parameter models are configured without introducing any mass, the foundation input motion can be directly applied to these models for their applications to the analysis of seismic excitation. The effectiveness of these new models is strictly validated by successfully simulating a semi-infinite bar on an elastic foundation. Subsequently, these models are applied for representing the dynamic stiffness functions for different types of foundation. Comparison of the new models with the other existing lumped-parameter models is also made to illustrate their advantages in requiring fewer parameters and featuring a more systematic expansion. Copyright © 2002 John Wiley & Sons, Ltd. [source] A parallel multigrid solver for high-frequency electromagnetic field analyses with small-scale PC clusterELECTRONICS & COMMUNICATIONS IN JAPAN, Issue 9 2008Kuniaki Yosui Abstract Finite element analyses of electromagnetic fields are commonly used for designing various electronic devices. The scale of the analyses becomes larger and larger, therefore, a fast linear solver is needed to solve linear equations arising from the finite element method. Since a multigrid solver is the fastest linear solver for these problems, parallelization of a multigrid solver is quite a useful approach. From the viewpoint of industrial applications, an effective usage of a small-scale PC cluster is important due to initial cost for introducing parallel computers. In this paper, a distributed parallel multigrid solver for a small-scale PC cluster is developed. In high-frequency electromagnetic analyses, a special block Gauss, Seidel smoother is used for the multigrid solver instead of general smoothers such as a Gauss, Seidel or Jacobi smoother in order to improve the convergence rate. The block multicolor ordering technique is applied to parallelize the smoother. A numerical example shows that a 3.7-fold speed-up in computational time and a 3.0-fold increase in the scale of the analysis were attained when the number of CPUs was increased from one to five. © 2009 Wiley Periodicals, Inc. Electron Comm Jpn, 91(9): 28, 36, 2008; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecj.10160 [source] Optimal Control of Rigid-Link Manipulators by Indirect MethodsGAMM - MITTEILUNGEN, Issue 1 2008Rainer Callies Abstract The present paper is a survey and research paper on the treatment of optimal control problems of rigid-link manipulators by indirect methods. Maximum Principle based approaches provide an excellent tool to calculate optimal reference trajectories for multi-link manipulators with high accuracy. Their major drawback was the need to explicitly formulate the complicated system of adjoint differential equations and to apply the full apparatus of optimal control theory. This is necessary in order to convert the optimal control problem into a piecewise defined, nonlinear multi-point boundary value problem. An accurate and efficient access to first- and higher-order derivatives is crucial. The approach described in this paper allows it to generate all the derivative information recursively and simultaneously with the recursive formulation of the equations of motion. Nonlinear state and control constraints are treated without any simplifications by transforming them into sequences of systems of linear equations. By these means, the modeling of the complete optimal control problem and the accompanying boundary value problem is automated to a great extent. The fast numerical solution is by the advanced multiple shooting method JANUS. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] Parsimonious finite-volume frequency-domain method for 2-D P,SV -wave modellingGEOPHYSICAL JOURNAL INTERNATIONAL, Issue 2 2008R. Brossier SUMMARY A new numerical technique for solving 2-D elastodynamic equations based on a finite-volume frequency-domain approach is proposed. This method has been developed as a tool to perform 2-D elastic frequency-domain full-waveform inversion. In this context, the system of linear equations that results from the discretization of the elastodynamic equations is solved with a direct solver, allowing efficient multiple-source simulations at the partial expense of the memory requirement. The discretization of the finite-volume approach is through triangles. Only fluxes with the required quantities are shared between the cells, relaxing the meshing conditions, as compared to finite-element methods. The free surface is described along the edges of the triangles, which can have different slopes. By applying a parsimonious strategy, the stress components are eliminated from the discrete equations and only the velocities are left as unknowns in the triangles. Together with the local support of the P0 finite-volume stencil, the parsimonious approach allows the minimizing of core memory requirements for the simulation. Efficient perfectly matched layer absorbing conditions have been designed for damping the waves around the grid. The numerical dispersion of this FV formulation is similar to that of O(,x2) staggered-grid finite-difference (FD) formulations when considering structured triangular meshes. The validation has been performed with analytical solutions of several canonical problems and with numerical solutions computed with a well-established FD time-domain method in heterogeneous media. In the presence of a free surface, the finite-volume method requires 10 triangles per wavelength for a flat topography, and fifteen triangles per wavelength for more complex shapes, well below the criteria required by the staircase approximation of O(,x2) FD methods. Comparisons between the frequency-domain finite-volume and the O(,x2) rotated FD methods also show that the former is faster and less memory demanding for a given accuracy level, an attractive feature for frequency-domain seismic inversion. We have thus developed an efficient method for 2-D P,SV -wave modelling on structured triangular meshes as a tool for frequency-domain full-waveform inversion. Further work is required to improve the accuracy of the method on unstructured meshes. [source] Modelling elastic media with the wavelet transformGEOPHYSICAL JOURNAL INTERNATIONAL, Issue 2 2001Joăo Willy Corręa Rosa Summary We present a new method for modelling 2-D elastic media with the application of the wavelet transform, which is also extended to cases where discontinuities simulate geological faults between two different elastic media. The basic method consists of the discretization of the polynomial expansion for the boundary conditions of the 2-D problem involving the stress and strain relations for the media. This parametrization leads to a system of linear equations that should be solved for the determination of the expansion coefficients, which are the model parameters, and their determination leads to the solution of the problem. The wavelet transform is applied with two main objectives, namely to decrease the error related to the truncation of the polynomial expansion and to make the system of linear equations more compact for computation. This is possible due to the properties of this finite length transform. The method proposed here was tested for six different cases for which the analytical solutions are known. In all tests considered, we obtained very good matches with the corresponding known analytical solutions, which validate the theoretical and computational parts of the project. We hope that the new method is useful for modelling real media. [source] Vertical vibration of an elastic strip footing on saturated soilINTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 5 2008Y. Q. Cai Abstract Based on Biot's dynamic coupled equations, the vertical vibration of an elastic strip footing on the surface of saturated soil is studied. Utilizing the Fourier transform, the governing dynamic differential equations for saturated poroelastic medium are solved. Considering the mixed boundary value conditions at the bottom of the foundation, a pair of dual integral equations about the vertical vibration of an elastic strip footing is derived, which can be converted to a set of linear equations by means of infinite series of orthogonal functions. The relation between the dynamic compliance coefficients and the dimensionless frequency tends to be gentle with decreasing footing rigidity, while the dimensionless frequency has only small effect on the dynamic compliance coefficients. When the dynamic permeability is large, its effect on the dynamic compliance coefficients should be taken into consideration. Furthermore, the dynamic compliance coefficients are found to be not sensitive to Poisson's ratio of the soil for footing on saturated soil. Copyright © 2007 John Wiley & Sons, Ltd. [source] Estimating spatial and parameter error in parameterized nonlinear reaction,diffusion equationsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 9 2007B. R. Carnes Abstract A new approach is proposed for the a posteriori error estimation of both global spatial and parameter error in parameterized nonlinear reaction,diffusion problems. The technique is based on linear equations relating the linearized spatial and parameter error to the weak residual. Computable local element error indicators are derived for local contributions to the global spatial and parameter error, along with corresponding global error indicators. The effectiveness of the error indicators is demonstrated using model problems for the case of regular points and simple turning points. In addition, a new turning point predictor and adaptive algorithm for accurately computing turning points are introduced. Copyright © 2006 John Wiley & Sons, Ltd. [source] A comparison of eigensolvers for large-scale 3D modal analysis using AMG-preconditioned iterative methodsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 2 2005Peter Arbenz Abstract The goal of our paper is to compare a number of algorithms for computing a large number of eigenvectors of the generalized symmetric eigenvalue problem arising from a modal analysis of elastic structures. The shift-invert Lanczos algorithm has emerged as the workhorse for the solution of this generalized eigenvalue problem; however, a sparse direct factorization is required for the resulting set of linear equations. Instead, our paper considers the use of preconditioned iterative methods. We present a brief review of available preconditioned eigensolvers followed by a numerical comparison on three problems using a scalable algebraic multigrid (AMG) preconditioner. Copyright © 2005 John Wiley & Sons, Ltd. [source] Fast direct solution of the Helmholtz equation with a perfectly matched layer or an absorbing boundary conditionINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 14 2003Erkki Heikkola Abstract We consider the efficient numerical solution of the Helmholtz equation in a rectangular domain with a perfectly matched layer (PML) or an absorbing boundary condition (ABC). Standard bilinear (trilinear) finite-element discretization on an orthogonal mesh leads to a separable system of linear equations for which we describe a cyclic reduction-type fast direct solver. We present numerical studies to estimate the reflection of waves caused by an absorbing boundary and a PML, and we optimize certain parameters of the layer to minimize the reflection. Copyright © 2003 John Wiley & Sons, Ltd. [source] A comparison of preconditioners for incompressible Navier,Stokes solversINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2008M. ur Rehman Abstract We consider solution methods for large systems of linear equations that arise from the finite element discretization of the incompressible Navier,Stokes equations. These systems are of the so-called saddle point type, which means that there is a large block of zeros on the main diagonal. To solve these types of systems efficiently, several block preconditioners have been published. These types of preconditioners require adaptation of standard finite element packages. The alternative is to apply a standard ILU preconditioner in combination with a suitable renumbering of unknowns. We introduce a reordering technique for the degrees of freedom that makes the application of ILU relatively fast. We compare the performance of this technique with some block preconditioners. The performance appears to depend on grid size, Reynolds number and quality of the mesh. For medium-sized problems, which are of practical interest, we show that the reordering technique is competitive with the block preconditioners. Its simple implementation makes it worthwhile to implement it in the standard finite element method software. Copyright © 2007 John Wiley & Sons, Ltd. [source] Splitting methods for high order solution of the incompressible Navier,Stokes equations in 3DINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 10-11 2005Arnim Brüger Abstract The incompressible Navier,Stokes equations are discretized in space by a hybrid method and integrated in time by the method of lines. The solution is determined on a staggered curvilinear grid in two space dimensions and by a Fourier expansion in the third dimension. The space derivatives are approximated by a compact finite difference scheme of fourth-order on the grid. The solution is advanced in time by a semi-implicit method. In each time step, systems of linear equations have to be solved for the velocity and the pressure. The iterations are split into one outer iteration and three inner iterations. The accuracy and efficiency of the method are demonstrated in a numerical experiment with rotated Poiseuille flow perturbed by Orr,Sommerfeld modes in a channel. Copyright © 2005 John Wiley & Sons, Ltd. [source] | ||||||||||||||||||||||||||||||||||||||||||||||