9. Determinants and matrices

2015 ◽  
pp. 111-125
Author(s):  
Peter M. Higgins
Keyword(s):  
Linear Transformation ◽  
Matrix Theory ◽  
Three Dimensional ◽  
Unit Cube ◽  
Three Dimensions ◽  
The Matrix ◽  
Dimensional Version ◽  
Key Topics ◽  
Cramer’S Rule ◽  
Kirchhoff Matrix

‘Determinants and matrices’ explains that in three dimensions, the absolute value of the determinant det(A) of a linear transformation represented by the matrix A is the multiplier of volume. The columns of A are the images of the position vectors of the sides of the unit cube and they define a three-dimensional version of a parallelogram, a parallelepiped, the volume of which is |det(A)|. It goes on to describe the properties and applications of determinants to networks (using the Kirchhoff matrix); Cramer’s Rule; eigenvalues; and eigenvectors, which are fundamental in linear mathematics. Other key topics in matrix theory—similarity, diagonalization, and factorization of matrices—are also discussed.

A generalized Archie’s law for n phases

Geophysics ◽  
2010 ◽  
Vol 75 (6) ◽  
pp. E247-E265 ◽  
Author(s):  
Paul W. J. Glover
Keyword(s):  
Volume Fraction ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Reservoir Rock ◽  
Archie’S Law ◽  
Phase Volume Fraction ◽  
General Law ◽  
Special Cases ◽  
The Matrix ◽  
Definition Of

Archie’s law has been the standard method for relating the conductivity of a clean reservoir rock to its porosity and the conductivity of its pore fluid for more than [Formula: see text]. However, it is applicable only when the matrix is nonconducting. A modified version that allows a conductive matrix was published in 2000. A generalized form of Archie’s law is studied for any number of phases for which the classical Archie’s law and modified Archie’s law for two phases are special cases. The generalized Archie’s law contains a phase conductivity, a phase volume fraction, and phase exponent for each of its [Formula: see text] phases. The connectedness of each of the phases is considered, and the principle of conservation of connectedness in a three-dimensional multiphase mixture is introduced. It is confirmed that the general law is formally the same as the classical Archie’s law and modified Archie’s law for one and two conducting phases, respectively. The classical second Archie’s law is compared with the generalized law, which leads to the definition of a saturation exponent for each phase. This process has enabled the derivation of relationships between the phase exponents and saturation exponents for each phase. The relationship between percolation theory and the generalized model is also considered. The generalized law is examined in detail for two and three phases and semiquantitatively for four phases. Unfortunately, the law in its most general form is very difficult to prove experimentally. Instead, numerical modeling in three dimensions is carried out to demonstrate that it behaves well for a system consisting of four interacting conducting phases.

Thermo-mechanical Effects on Fracture Growth and Apertures in Three-Dimensional Subsurface Fractured Rocks 

2021 ◽  
Author(s):  
Adriana Paluszny ◽  
Robin N. Thomas ◽  
M. Cristina Saceanu ◽  
Robert W. Zimmerman
Keyword(s):  
Heat Transfer ◽  
Fluid Flow ◽  
Three Dimensional ◽  
Growth Patterns ◽  
Three Dimensions ◽  
Interaction Patterns ◽  
Flow Equations ◽  
The Matrix ◽  
Model Equations ◽  
Fracture Growth

<p>A finite-element based, quasi-static growth algorithm models mixed mode concurrent fracture growth in three dimensions, leading to the formation of non-planar arrays and networks. To model the fully coupled THM model, equations describing mechanical deformation as well as heat transfer in the matrix and in the fractures are introduced in the formulation, simultaneously accounting for the effect of fluid flow and stress-strain response. This results in five separate, but two-way coupled model equations: a thermoporoelastic mechanical model; two fluid flow equations, one for the rock matrix and one for the fractures; two heat transfer equations, similarly for both the matrix and fractures. Fractures are represented explicitly as discrete surfaces embedded within a volumetric domain [1]. Growth is computed as a set of vectors that modify the geometry of a fracture by accruing new fracture surfaces in response to brittle deformation. Fracture tip stress intensity factors drive fracture growth. This growth methodology is validated against analytical solutions for fractures under compression and tension [2]. Thermal effects on the apertures and growth patterns will be presented. Isolated fracture geometries are compared with selected experimental results on brittle media. Accurate growth is demonstrated for domains discretised by refined and coarse volumetric meshes. Fracture and volume-based growth rates are shown to modify fracture interaction patterns. Two-dimensional cut-plane views of fracture networks show how fractures would appear on the surface of the studied volume.</p><p><strong>REFERENCES</strong></p><p>[1] N. Thomas, A. Paluszny and R. W. Zimmerman. Growth of three-dimensional fractures, arrays, and networks in brittle rocks under tension and compression. Computers and Geotechnics, 2020. doi: 10.1016/j.compgeo.2020.103447</p><p>[2] Paluszny and R. W. Zimmerman. Numerical fracture growth modeling using smooth surface geometric deformation. Eng. Fract. Mech., 108, 19-36, 2013. doi: 10.1016/j.engfracmech.2013.04.012</p>

Interphase effects on the thermo-mechanical properties of three-phase composites

2016 ◽  
Vol 230 (19) ◽  
pp. 3361-3371 ◽  
Author(s):  
Mohammad K Hassanzadeh-Aghdam ◽  
Mohammad J Mahmoodi ◽  
Reza Ansari
Keyword(s):  
Mechanical Properties ◽  
Three Dimensional ◽  
Longitudinal Direction ◽  
Fiber Composites ◽  
Volume Element ◽  
Three Dimensions ◽  
Spherical Particles ◽  
Aspect Ratios ◽  
Three Phase ◽  
The Matrix

A three-dimensional micromechanics-based analytical model is developed to investigate the influence of interphase on the thermo-mechanical properties of three-phase composites. The representative volume element (RVE) of composites is extended to c × r × h cells in three dimensions and the RVE consists of three phases including filler, matrix and interphase. The arrangement state of filler within the matrix materials is assumed to be random with uniform distribution. Fillers are surrounded by the interphase in the whole composite. The effects of interphase such as its thickness and stiffness on the thermo-mechanical properties of composite with various aspect ratios of filler are studied. The results illustrate that while the effects of interphase is significant for composites with randomly distributed spherical particles, it turns to be less effective as the aspect ratio of filler of composite increases. Moreover, the results demonstrate that the effect of interphase on the thermo-mechanical properties of fibrous composites in the transverse direction is more significant than that of fiber composites in the longitudinal direction.

Random matrix theory and ribonucleic acid (RNA) folding

2018 ◽  
Author(s):  
Antonia Tulino ◽  
Sergio Verdu
Keyword(s):  
Random Matrix Theory ◽  
Physical Interpretation ◽  
Random Matrix ◽  
Mechanical Model ◽  
Rna Folding ◽  
Matrix Theory ◽  
Three Dimensions ◽  
Large N ◽  
Rna Pseudoknots ◽  
The Matrix

This article discusses a series of recent applications of random matrix theory (RMT) to the problem of RNA folding. It first provides a schematic overview of the RNA folding problem, focusing on the concept of RNA pseudoknots, before considering a simplified framework for describing the folding of an RNA molecule; this framework is given by the statistic mechanical model of a polymer chain of L nucleotides in three dimensions with interacting monomers. The article proceeds by presenting a physical interpretation of the RNA matrix model and analysing the large-N expansion of the matrix integral, along with the pseudoknotted homopolymer chain. It extends previous results about the asymptotic distribution of pseudoknots of a phantom homopolymer chain in the limit of large chain length.

Fine-Scale Simulation of Sandstone Acidizing

2005 ◽  
Vol 127 (3) ◽  
pp. 225-232 ◽  
Author(s):  
Chunlou Li ◽  
Tao Xie ◽  
Maysam Pournik ◽  
Ding Zhu ◽  
A. D. Hill
Keyword(s):  
Flow Field ◽  
Three Dimensional ◽  
Axial Direction ◽  
Three Dimensions ◽  
Scale Model ◽  
Small Scale ◽  
Fine Scale ◽  
Homogeneous Case ◽  
The Matrix ◽  
Highly Correlated

We have developed a fine-scale model of the sandstone core acid flooding process by solving acid and mineral balance equations for a fully three-dimensional flow field that changed as acidizing proceeded. The initial porosity and mineralogy field could be generated in a correlated manner in three dimensions; thus, a laminated sandstone could be simulated. The model has been used to simulate sandstone acidizing coreflood conditions, with a 1in.diam by 2in. long core represented by 8000 grid blocks, each having different initial properties. Results from this model show that the presence of small-scale heterogeneities in a sandstone has a dramatic impact on the acidizing process. Flow field heterogeneities cause acid to penetrate much farther into the formation than would occur if the rock were homogeneous, as is assumed by standard models. When the porosity was randomly distributed (sampled from a normal distribution), the acid penetrated up to twice as fast as in the homogeneous case. When the porosity field is highly correlated in the axial direction, which represents a laminated structure, acid penetrates very rapidly into the matrix along the high-permeability streaks, reaching the end of the simulated core as much as 17 times faster than for a homogeneous case.

scholarly journals Evaluating secondary inorganic aerosols in three dimensions

2016 ◽  
Vol 16 (16) ◽  
pp. 10651-10669 ◽  
Author(s):  
Keren Mezuman ◽  
Susanne E. Bauer ◽  
Kostas Tsigaridis
Keyword(s):  
Nitric Acid ◽  
Climate Model ◽  
Three Dimensional ◽  
Model Performance ◽  
Heterogeneous Reactions ◽  
Three Dimensions ◽  
Surface Measurement ◽  
The Matrix ◽  
The Usa ◽  
Dimensional Distribution

Abstract. The spatial distribution of aerosols and their chemical composition dictates whether aerosols have a cooling or a warming effect on the climate system. Hence, properly modeling the three-dimensional distribution of aerosols is a crucial step for coherent climate simulations. Since surface measurement networks only give 2-D data, and most satellites supply integrated column information, it is thus important to integrate aircraft measurements in climate model evaluations. In this study, the vertical distribution of secondary inorganic aerosol (i.e., sulfate, ammonium, and nitrate) is evaluated against a collection of 14 AMS flight campaigns and surface measurements from 2000 to 2010 in the USA and Europe. GISS ModelE2 is used with multiple aerosol microphysics (MATRIX, OMA) and thermodynamic (ISORROPIA II, EQSAM) configurations. Our results show that the MATRIX microphysical scheme improves the model performance for sulfate, but that there is a systematic underestimation of ammonium and nitrate over the USA and Europe in all model configurations. In terms of gaseous precursors, nitric acid concentrations are largely underestimated at the surface while overestimated in the higher levels of the model. Heterogeneous reactions on dust surfaces are an important sink for nitric acid, even high in the troposphere. At high altitudes, nitrate formation is calculated to be ammonia limited. The underestimation of ammonium and nitrate in polluted regions is most likely caused by a too simplified treatment of the NH3 ∕ NH4+ partitioning which affects the HNO3 ∕ NO3− partitioning.

convergent-beam diffraction from surfaces

1987 ◽  
Vol 45 ◽  
pp. 30-33
Author(s):  
J. A. Eades ◽  
A. E. Smith ◽  
D. F. Lynch
Keyword(s):  
Reciprocal Lattice ◽  
Three Dimensional ◽  
Grazing Incidence ◽  
Three Dimensions ◽  
Plane Figure ◽  
Transmission Electron ◽  
Convergent Beam ◽  
Beam Patterns ◽  
Beam Diffraction

It is quite simple (in the transmission electron microscope) to obtain convergent-beam patterns from the surface of a bulk crystal. The beam is focussed onto the surface at near grazing incidence (figure 1) and if the surface is flat the appropriate pattern is obtained in the diffraction plane (figure 2). Such patterns are potentially valuable for the characterization of surfaces just as normal convergent-beam patterns are valuable for the characterization of crystals.There are, however, several important ways in which reflection diffraction from surfaces differs from the more familiar electron diffraction in transmission.GeometryIn reflection diffraction, because of the surface, it is not possible to describe the specimen as periodic in three dimensions, nor is it possible to associate diffraction with a conventional three-dimensional reciprocal lattice.

Designing TIMP (tissue inhibitor of metalloproteinases) variants that are selective metalloproteinase inhibitors

2003 ◽  
Vol 70 ◽  
pp. 201-212 ◽  
Author(s):  
Hideaki Nagase ◽  
Keith Brew
Keyword(s):  
Three Dimensional ◽  
Therapeutic Interventions ◽  
Tissue Inhibitors Of Metalloproteinases ◽  
Structural Basis ◽  
Structural Elements ◽  
Ridge Structure ◽  
Extracellular Matrix Components ◽  
Metalloproteinase Inhibitors ◽  
The Matrix ◽  
A Disintegrin And Metalloproteinase

The tissue inhibitors of metalloproteinases (TIMPs) are endogenous inhibitors of the matrix metalloproteinases (MMPs), enzymes that play central roles in the degradation of extracellular matrix components. The balance between MMPs and TIMPs is important in the maintenance of tissues, and its disruption affects tissue homoeostasis. Four related TIMPs (TIMP-1 to TIMP-4) can each form a complex with MMPs in a 1:1 stoichiometry with high affinity, but their inhibitory activities towards different MMPs are not particularly selective. The three-dimensional structures of TIMP-MMP complexes reveal that TIMPs have an extended ridge structure that slots into the active site of MMPs. Mutation of three separate residues in the ridge, at positions 2, 4 and 68 in the amino acid sequence of the N-terminal inhibitory domain of TIMP-1 (N-TIMP-1), separately and in combination has produced N-TIMP-1 variants with higher binding affinity and specificity for individual MMPs. TIMP-3 is unique in that it inhibits not only MMPs, but also several ADAM (a disintegrin and metalloproteinase) and ADAMTS (ADAM with thrombospondin motifs) metalloproteinases. Inhibition of the latter groups of metalloproteinases, as exemplified with ADAMTS-4 (aggrecanase 1), requires additional structural elements in TIMP-3 that have not yet been identified. Knowledge of the structural basis of the inhibitory action of TIMPs will facilitate the design of selective TIMP variants for investigating the biological roles of specific MMPs and for developing therapeutic interventions for MMP-associated diseases.

Extension of the Spatial PDI Model to Three Dimensions

1997 ◽  
Vol 84 (1) ◽  
pp. 176-178
Author(s):  
Frank O'Brien
Keyword(s):  
Population Density ◽  
Dimensional Space ◽  
Finite Lattice ◽  
Three Dimensional ◽  
Upper Bounds ◽  
Three Dimensions ◽  
Lower And Upper Bounds ◽  
Density Index ◽  
Three Dimensional Space

The author's population density index ( PDI) model is extended to three-dimensional distributions. A derived formula is presented that allows for the calculation of the lower and upper bounds of density in three-dimensional space for any finite lattice.

3D simulation of emulsion flow using the boundary element method on the heterogeneous computational systems

2012 ◽  
Vol 9 (1) ◽  
pp. 142-146
Author(s):  
O.A. Solnyshkina
Keyword(s):  
Boundary Element Method ◽  
Boundary Element ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Problem Size ◽  
Low Reynolds Numbers ◽  
Infinite Domains ◽  
The Matrix ◽  
Computational Systems ◽  
Element Method

In this work the 3D dynamics of two immiscible liquids in unbounded domain at low Reynolds numbers is considered. The numerical method is based on the boundary element method, which is very efficient for simulation of the three-dimensional problems in infinite domains. To accelerate calculations and increase the problem size, a heterogeneous approach to parallelization of the computations on the central (CPU) and graphics (GPU) processors is applied. To accelerate the iterative solver (GMRES) and overcome the limitations associated with the size of the memory of the computation system, the software component of the matrix-vector product

Sign in / Sign up

Export Citation Format

Share Document