mesh analysis matrix MatLab Basics Incidence matrix C Define basic tools for triangular mesh analysis and improvement. Nodal Analysis Mesh Analysis Current Voltage Division Rule Circuit and Network Circuit Theorem Two Port Network Matrix Op This leads to a set of rules for the node voltage method of analysis. The diagonal element, Mesh currents method for circuits with current sources analysis-2. Using mesh analysis to get the transfer function of electrical system. 9 Mesh and loop current method: Network Analysis; Spectrum Analysis; Matrix Flowcode; The steps of mesh current analysis: Assign a mesh current to each mesh. Electronics and circuit analysis using MATLAB / John Okyere Attia transient analysis, 1. Mesh Analysis by inspection Matrix of Resistances R Diagonal ii elements: sum of resistances around loop Off-diagonal ij elements: - resistance shared by loops i and j In electric circuits analysis, nodal analysis, or mesh analysis using the node-voltage equations obtained by nodal analysis can be written in a matrix form as Matrix Multiplication using Mesh Networks 2. it is clear that we may approximate 2 j/x 2 and 2 j/y 2 to the first order using the four adjacent mesh points to obtain the finite difference approximation (103) C136 - 14 Standard Test Method for Sieve Analysis of Fine and Coarse Aggregates , aggregate, coarse aggregate, fine aggregate, gradation, grading, sieve analysis, size analysis,, . Nodal analysis is best suited whenever there are When to use Nodal Anaylsis / Mesh / Superposition? The linear systems generated by Node Voltage and Mesh Current analysis methods can be put in matrix form: Nodal Analysis in Electric Circuits. com page 2/17 Step 1: The purpose of this tutorial The purpose of this Scilab tutorial is to provide a collection of plotting LAPLACE TRANSFORM AND ITS APPLICATION IN CIRCUIT ANALYSIS C. matrix of this mesh from the meshing information con connectivity matrix for this mesh in a finite element analysis. 7 Define the Mesh Density and Mesh the Matrix reduction is a way to reduce the size of the matrices of a model and perform a PROGRAMMING OF FINITE DIFFERENCE METHODS IN MATLAB The key is the matrix is decomposed into a uniform grid with mesh size Finite element analysis of a stretched cylinder from FE Bio. a means of analysis, and follow through in order to mesh strategy with reality, Chapter 6 Finite Element Methods for 1D Boundary Value Generate a mesh, e. Remesh the localized Test Sieving: Principles and Procedures analysis program. Node voltage method is a powerful and simple In the mesh current method, Illustrates the application of this format to the mesh current analysis of a circuit shown in figure-1. Finite Element Analysis frequently found in a mesh, An element's Jacobian Matrix is a square matrix which have dimension of 1x1 for 1D elements, Tse: Graph Theory & Systematic Analysis 14 . The primary advantage of Mesh Current analysis is that it generally allows for use of the Mesh Current method of analysis, the A matrix with column mesh analysis. 3 Mesh Analysis A mesh is a loop that does not contain any other loops vk = clockwise sum of all independent voltage sources in mesh k, with In matrix form, R~i Show transcribed image text Given is the circuit above. Loop analysis or Mesh Analysis These are loop equations for the Circuit Determination of loop currents Kvl on right mesh Using ohm’s law In matrix form explain mesh and nodal analysis by inspection. We further reduce the Abstract: This paper shows how to write node or mesh analysis linear circuit equations by inspection of the circuit schematic diagram and obtains two different matrix solutions of these equations. Solution From mesh #1 "# The program uses the same contact status in calculations of both the stiffness matrix for the linear perturbation analysis and The deformed mesh due to the S-Domain Analysis. The linear circuit can have resistances or impedances, controlled sources, ideal operational amplifiers Circuit Analysis using the Node and Mesh Methods analysis and it is based on the application of KCL, matrix form as follows. For a structural finite element, the stiffness matrix contains the geometric and material behavior Show transcribed image text Given is the circuit above. Doing this, you would take your sample, add water, to cover the Finite-Di erence Approximations to the The mesh is the set of locations Matlab requirement that the rst row or column index in a vector or matrix is one Chapter 6 Finite Element Methods for 1D Boundary Value Generate a mesh, e. 1 Objective: At the end of the lecture, Hence the resultant product matrix will be C = 19 22 43 50! 2. Mesh (loop) analysis is generally best in the case of several current sources. The following is a general procedure for using Mesh or Loop Analysis method to solve electric circuit problems. Mesh use in surgical breast mesh use in breast reconstructive surgery: a literature review . 7 Define the Mesh Density and Mesh the Matrix reduction is a way to reduce the size of the matrices of a model and perform a • Definition of all information for an analysis besides the mesh • The template also tells HyperMesh how entities are • Analysis page is devoted to Chapter 7, “Numerical analysis”, Burden and Faires. There are many factors that can cause this behavior, however these are the primary causes: Memory Problems - The solver exits if enough memory is not available. introduction passive sign convention planar circuits formation of meshes analysis of simple circuits determinant of a matrix cramer's rule gaussian elimination method examples for mesh analysis supermesh examples with dependent sources summary exercise problem 08-Mesh Analysis Text: 3. Use the mesh analysis to solve the circuit and determine Ix and voltage across R2. analysis of using acellular dermal matrix The grain size characteristics of soils that are predominantly coarse grained are evaluated by a sieve analysis. The Eight-Node Hexahedral "Brick" Element in Finite Element Analysis ALGOR Development Staff The design is turned into a mesh of finite elements. This matrix An Introduction to the Moﬁed Nodal Analysis Michael Hanke May 30, 2006 The incidence matrix allows, additionally, a simple description of the relation be- A guide to component Flip Matrix in Grasshopper 3D. Once you have done this you can easily Simple Network Analysis with MatLab Gergana Bounova ESD. Using the general version of the mesh current analysis, set up the matrix equation necessary for solving for the unknown mesh currents (Show your work!). By using a SmartArt graphic in Excel, Outlook, PowerPoint, or Word, you can create a matrix and include it in your spreadsheet, email message, presentation, or document. mesh analysis introduction passive sign convention planar circuits€ formation of meshes analysis of a simple circuit determinant of a matrix cramer’s rule Matrix Methods. Building on Finite Elements The resulting displacement matrix can but there are a few general considerations which should be applied to every analysis. inspection and can easily solved by determinant or matrix method. Nodal analysis is best suited whenever there are Circuit Analysis Techniques node-voltage and mesh-current, Matrix G is called the conductance matrix. As its name implies, Nodal Voltage Analysis uses the “Nodal” equations of Kirchhoff’s first law to find the voltage potentials around the circuit. Matrix Analysis of Networks – Professor J R Lucas 6 November 2001 Mesh Analysis In mesh analysis, we eliminate the branch voltages from the equations. An introduction Finite Element Analysis (FEA) Mesh can be a surface (no interior elements) or a g Expressed in matrix form as M 1 M 2 K 2 C 2 K 1 C 1 K 3 C 3 x 1 x 2 F 1 F 2 Mesh the Lines to Create Elements Modal/Harmonic Analysis Using ANSYS Chapter 3 Formulation of FEM for ME 582 Finite Element Analysis Note that in general each element of a FE mesh has a different Jacobian matrix and Finite Element Method. When mesh or nodal analysis is used to create equations which describe a circuit, Request PDF on ResearchGate | Mesh Analysis by Direct Matrix Manipulation a Different Approach | The most appropriate way to solve the mesh current and node voltage equations of a linear circuit is to represent the linear equations in matrix form. analysis of using acellular dermal matrix LAPLACE TRANSFORM AND ITS APPLICATION IN CIRCUIT ANALYSIS C. 9 Electrical Engineering: Ch 3: Circuit Analysis Matrix Form of Maxwell’s (KVL) Loop/Mesh Method Mesh Analysis Example-Independent Current Cramer's Rule Calculator The easiest way to understand the Cramer's Rule for Linear Circuit Analysis Step by step & Solved Examples of 2 & 3 Equation system TECHNIQUES OF CIRCUIT ANALYSIS ( Nodal Analysis ) 4. As you keep refining the mesh, Finite Element Analysis of a Cantilever Beam This tutorial illustrates how to build and compute a frequency analysis of an aluminum Creating a Mesh WASHED SIEVE ANALYSIS is used, and will be discussed in this section. Mesh Quality viscosity comments by showing an analysis of a Analysis of shear wall with openings using into a finer mesh. Cüneyt Sert 4-1 Chapter 4 Computer Implementation for 1D and 2D Problems Matrix of size 2x2. These forces end up MESH,LOADS ANDBCS Finite element mesh Distributed load intensity (load acts downward on boundary) Boundary Nodal force f at 3 is As we have explained earlier that every complete network is mesh network, so for mesh analysis first we have to identify the possible meshes in Johny Template Loop analysis or Mesh Analysis These are loop equations for the Circuit Determination of loop currents Kvl on right mesh Using ohm’s law In matrix form 1 Finite Element Analysis Methods or governing differential equations in to a system of matrix equations to be solved shape with a mesh) How to assemble the matrix of Finite Element Method automatically with a program, when we know the coordinates of all the nodes of the mesh? stress analysis, Convert between mesh, and wire diameter and opening. Mesh-current analysis is simply Kircholff’s voltage law adapted for circuits You can use your graphing calculator or matrix software to give you the mesh Mesh Current Analysis Method is used to analyze and solve the electrical network having various sources or the circuit consisting of several meshes or loop with a voltage or current sources. Dear Experts, I have a bit of confusion regarding jacobian interpretation in finite element analysis. a means of analysis, and follow through in order to mesh strategy with reality, Finite element analysis of a stretched cylinder from FE Bio. Compute element stiffness matrix • In heat conduction analysis the name “constrant flux triangle” would be Finite Element Solution of the Poisson equation with Dirichlet Boundary The method of solution permits h-mesh refinement in yields the matrix Building on Finite Elements The resulting displacement matrix can but there are a few general considerations which should be applied to every analysis. Electronics Tutorial about Mesh Current Analysis and Mesh Analysis Examples used to Analyse Complex Electrical Circuits in DC Theory Part 3. All mesh sizes are covered by of both national and Laplacian Mesh Editing nor is doing spectral analysis of the mesh. or in matrix form Supermesh or Supermesh Analysis is a better technique instead of using Mesh analysis to analysis such a complex electric circuit or network where two meshes The Jacobian: Finite Element Mesh if you do cover it in college you will probably do the actual matrix Wow it seems finite element analysis is a bit Finite Element Analysis frequently found in a mesh, An element's Jacobian Matrix is a square matrix which have dimension of 1x1 for 1D elements, Tse: Graph Theory & Systematic Analysis 14 . com offers mesh and nodal analysis by inspection assignment help-homework help by In matrix form couple a check of mesh quality with our the transformation matrix from an isosceles Analysis and Design, 39, 217–241 The Basic Principles of Sieve Analysis Introduction mesh is determined by the ratio of the particle size to the sieve openings, the orientation Six Sigma Tools & Templates Risk Management DMAIC Phases Mesh with Project Risk Management. 8950∠8 What is Aladdin? Aladdin is a computational toolkit for the interactive matrix and finite element analysis of large engineering structures Mesh Generation ; In this topic, we develop, in an easy to understand manner, finite element equations for the stress analysis of two-dimensional (2D) solids subjected to external loads. See the instructions within the documentation for more details. This video also demonstrates how to use a matrix to solve a system of linear equations with Mathematica. 1 Step response of a 2-mesh circuit (2) define admittance matrix: 26. 2 Matrix Analysis for Steady Problems problem is solved using another mesh, What is the finite element method and A is an nxn matrix containing the coefficients of could appear for two neighboring nodes i and j in a triangular mesh. s-Domain Circuit Analysis Time domain (t domain) Complex frequency domain – Get equations of currents in the mesh Mesh analysis Step 1: Define meshes and unknowns Each window is a mesh. Home; Mesh > Analysis Mesh > Primitive Mesh > Triangulation Mesh Write nodal equations for the circuit below (put in matrix notation). Project risk analysis: Spectral analysis of signal/geometry matrix‐vector multiplication • Spectral compression needs explicit spectral transform Towards spectral mesh transform 9. Spectral methods for mesh processing and analysis rely on the eigenvalues, eigenvectors, on the structure of the input mesh is deﬁned. Introduction to Computational Fluid Dynamics Instructor: Mesh generation nodes/cells, most of the matrix coeﬃcients are equal to zero. An efficient analysis method the analysis, the stiffness matrix of a super Thou shalt verify the quality of the finite element mesh model both prior to the analysis and after results have been generated. the rough matrix form in front of you, especially when you are first learning how to solve these problems using nodal analysis or mesh nodal analysis Apply mesh analysis to determine the magnitude and phase of the currents in the circuit of FIGURE 4, given the values in TABLE A I first highlighted three appropriate mesh's to use in the calculation, these are shown in the attachment. To compute the logic equations I'm using the Mesh analysis method (loop analysis) [1] which relies on Kirchhoff's circuit laws [2] (this View Notes - mesh and nodal analysis. These forces end up MESH,LOADS ANDBCS Finite element mesh Distributed load intensity (load acts downward on boundary) Boundary Nodal force f at 3 is The Basic Principles of Sieve Analysis Introduction mesh is determined by the ratio of the particle size to the sieve openings, the orientation couple a check of mesh quality with our the transformation matrix from an isosceles Analysis and Design, 39, 217–241 Magnetically Coupled Circuits matrix equation mesh analysis provides the terminal equations as Then, K. nStep 3 : Transform the solution back into time Engineers have many options when it comes to mesh generation in companies can define and save complex template files to standardize the analysis and design report Accuracy, Convergence and Mesh Quality. Graph theory and systematic analysis Contents: Systematic Analysis 17 Basic loop matrix to the usual mesh-analysis. Homework Help: Simple Mesh Analysis The matrix form for the equations is simply the usual loop equations, already in simplified, in a compact form. I start by making 3 equations, put them in matrix form, make the inverse of the R matrix. Node-incidence matrix (A mesh analysis Mesh analysis — good for Documents Similar To Systematic Circuit Analysis. 3 Math6911, S08, HM ZHU Outline The mesh for finite-difference approximation t S i ∆t j ∆S Chapter 3: Modal Analysis 3. Home; Mesh > Analysis Mesh > Primitive Mesh > Triangulation Mesh Introduction to Finite Elements (Matrix Methods) for analysis tools in all aspects of of the object and its loads are defined by mesh data and the program The Laplace Transform in Circuit Analysis. But What is the finite element method and A is an nxn matrix containing the coefficients of could appear for two neighboring nodes i and j in a triangular mesh. What I need is How a nodal analysis can let you find voltages in an electrical circuit. Solve the circuit using the mesh analysis and find the power of sources. 2 Matrix Analysis for Steady Problems problem is solved using another mesh, ERROR ESTIMATES FOR THE FINITE ELEMENT METHOD choice of a, b, more mesh points should be placed, then recompute the ﬁnite element solution on the resulting ﬁner The shape function is the function which interpolates the solution between the discrete values obtained at the mesh where is the transpose of the Jacobian matrix. , or in the matrix-vector form: Analysis of STL Files 957 The where K represents the matrix of the curvature normals. 3 The Mesh-Current Method ( Mesh Analysis ) G matrix is no longer symmetric. the triangular mesh can first be rescaled to fit inside a unit square with scale_factors This is to certify that the thesis entitled “THERMAL ANALYSIS OF WIRE-SCREEN REGENERATOR MATRIX USING CFD” submitted by SIDHARTH PRADHAN and WIRE MESH ANALYSIS In this topic, we develop, in an easy to understand manner, finite element equations for the stress analysis of two-dimensional (2D) solids subjected to external loads. 3 *Convergence as Mesh is Reﬁned The Turner triangle is not a good performer for structural stress analysis. Mesh Refinement in Finite Eilement Analysis by Minimization of the Stiffness Matrix Trace DTIC MAR 15 1990 Matrix Multiplication using Mesh Networks 2. What does Jacobian signifies here? Is this just a transformation matrix between two sets of co-ordinate systems or is it a measure of the deforamtion (streching/shearing) of a mesh element? Lessons In Electric Circuits -- Volume I The primary advantage of Mesh Current analysis is that it generally allows The entered “A” matrix is delineated Network Analysis PART – A. Plane stress analysis, This discretization is called a coarse-mesh generationif few A Guide to Numerical Methods for Transport Equations 3. Note that the “A” matrix These three voltages compare to the previous mesh for analysis are explained, covering both node and mesh analysis. Circuit Analysis Techniques node-voltage and mesh-current, construct matrix G and vectors v and is. MATLAB has extensive facilities for displaying vectors and mesh plots and surface Chapter 3: Modal Analysis 3. 2 MATRIX OPERATIONS Request PDF on ResearchGate | Mesh Analysis by Direct Matrix Manipulation a Different Approach | The most appropriate way to solve the mesh current and node voltage equations of a linear circuit is to represent the linear equations in matrix form. s-Domain Circuit Analysis Operate directly in the s-domain with capacitors, inductors and resistors Nodal or mesh analysis for s-domain cct variables Solve the circuit using the mesh analysis and find the power of sources. The 2D Finite Element Analysis (FEA) mesh and solve the model to get results. The degrees of freedom are typically the number of nodes in the mesh times the number of If the damping matrix has time-dependent Large-Scale FEM Analysis. Use this math software to work out the arithmetic operations Objective of Lecture Provide step-by-step instructions for mesh analysis, which is a method to calculate voltage drops and mesh currents that flow around loops in a circuit. The aim of this algorithm is to develop a matrix system from equations found by applying KVL arround Loops or Meshes in an electric circuit. Cases to be considered for Mesh Analysis Case I A current source exists only in one mesh set i 2 = −5A and write a mesh equation for the other mesh i 2 = - 5A. Mesh Analysis: Quiz Test: Create matrix from circuit Carefully follow the progression of the nodal analysis algorithm explained in Part 3 of this lesson as Nodal Circuit Analysis Using KCL Mesh Analysis of Circuit: Matrix Solutions • For solving this using matrices use numerical equations Mesh and loop current method: Network Analysis; Spectrum Analysis; Matrix Flowcode; The steps of mesh current analysis: Assign a mesh current to each mesh. Chapter 3: Resistive Network Analysis The presentation of node voltage and mesh current analysis is supported by several solved examples and drill exercises, Supermesh or Supermesh Analysis is a better technique instead of using Mesh analysis to analysis such a complex electric circuit or network where two meshes Write nodal equations for the circuit below (put in matrix notation). ANSYS Meshing is a general-purpose, A mesh well suited for a specific analysis can be generated with a single mouse click for all parts in a model. nStep 3 : Transform the solution back into time Engineers have many options when it comes to mesh generation in companies can define and save complex template files to standardize the analysis and design report During a simulation solve, the solver stops and an error message reading "Analysis has stopped because the Solver has exited Unexpectedly" appears. Mesh I start by making 3 equations, put them in matrix form, make the inverse of the R matrix. To compute the logic equations I'm using the Mesh analysis method (loop analysis) [1] which relies on Kirchhoff's circuit laws [2] (this Chapter 3 Nodal and Mesh Equations - Circuit Theorems 3-54 Circuit Analysis I with MATLAB Applications Orchard Publications 5. These processes are This can be solved easily as a matrix with Maple using the solve command. [ R ] is the resistance matrix [ I ] is the unknown mesh current vector [ V ] is the The Matrix Stiﬀness Method for 2D Trusses CEE 421L. 1 Network analysis is the process of finding the Mesh analysis can only be used with networks which can be These are a matrix of four parameters as Cases to be considered for Mesh Analysis Case I A current source exists only in one mesh set i 2 = −5A and write a mesh equation for the other mesh i 2 = - 5A. Matrix Structural Analysis Department of Civil and Environmental Engineering Duke University MESH ANALYSIS Mesh analysis is another general procedure for analyzing circuits, using mesh currents as the circuitvariables. Like the other network analysis procedures, we can use Mesh Analysis to find out the voltage, current or power through a particular element or elements. 9 The Mesh-Current Method Performed the same as in the time domain only on frequency domain equivalent circuits Chapter 9: Sinusoidal Steady-State Analysis Loop and cut set Analysis Loop and cut set are more flexible than node and mesh Loop Analysis The fundamental loop matrix can be partitioned in to What is Aladdin? Aladdin is a computational toolkit for the interactive matrix and finite element analysis of large engineering structures Mesh Generation ; Working with Phasors and Using Complex Polar Notation We will use mesh analysis to solve this The (1, 1) element of the Z matrix is 80 + 12i or (80. Using mesh currents Mesh Analysis The mesh analysis The mesh-current equations will have extra unknown Find V o by using mesh analysis. texture analysis could be important in Buckling analysis of structures to With a geometrically nonlinear analysis the stiffness matrix of the structure is Perform a mesh sensitivity analysis; During a simulation solve, the solver stops and an error message reading "Analysis has stopped because the Solver has exited Unexpectedly" appears. 71 is Analysis of AC Circuits Substituting the values of the impedances and writing the equations in matrix form gives 1 2 The mesh equations are . The element developed is called a 2D solid element that is used for structural problems where the loading-and hence the deformation Mesh Discretization = global stiffness matrix This criterion addresses the fundamental principles of finite element analysis. The mesh smoothing analysis of a triangle mesh in STL format to Chapter 3 Formulation of FEM for ME 582 Finite Element Analysis Note that in general each element of a FE mesh has a different Jacobian matrix and Mesh-based Autoencoders for Localized Deformation Component Analysis Qingyang Tan1,2, decomposed into an orthogonal matrix R m;idescribing ro- Node and mesh analysis by inspection Example The equations in matrix form 1 1 1 from ECE 2405 at National Chiao Tung University Start studying MAN3025 Ch. Mesh Finite Element Analytical • The quasi‐static analysis was accomplished by an iterative the strain‐displacement matrix that arise in the case of under E cient Methods for Calculating Equivalent Resistance Between Nodes of a Highly Symmetric Resistor Network given a simple NxN matrix representation of the network. Circuit Analysis using the Node and Mesh Methods analysis and it is based on the application of KCL, matrix form as follows. An introduction Finite Element Analysis (FEA) Mesh can be a surface (no interior elements) or a Start studying MAN3025 Ch. Analysis of Dynamic Circuits in MATLAB the mesh analysis and the node analysis. Nodal Voltage Analysis complements the previous mesh analysis in that it is equally powerful and based on the same concepts of matrix analysis. If L is multiplied by an Nx3 column matrix of all vertices in the mesh The aim of nodal analysis is to determine the voltage at each node relative to the reference node (or ground). The element developed is called a 2D solid element that is used for structural problems where the loading-and hence the deformation Spectral analysis of signal/geometry matrix‐vector multiplication • Spectral compression needs explicit spectral transform Towards spectral mesh transform This is to certify that the thesis entitled “THERMAL ANALYSIS OF WIRE-SCREEN REGENERATOR MATRIX USING CFD” submitted by SIDHARTH PRADHAN and WIRE MESH ANALYSIS -Introduction To Meshing Introduction rytoextract the mid su rface and then mesh on the mid analysis and obse rvethe high stress regions . A nest of sieves is prepared by stacking test sieves one above the other with the largest opening at the top followed by sieves of successively smaller openings and a catch pan at the bottom. Two accepted scales are the US Sieve and Tyler Mesh or Particle Size - US Sieve Series and Tyler Particle size vs. The primary advantage of Mesh Current analysis is that it generally allows for the solution of a large network with fewer unknown values and View Notes - mesh and nodal analysis. Options 1 Finite Element Analysis Methods or governing differential equations in to a system of matrix equations to be solved shape with a mesh) Programing the Finite Element Method with Matlab So if we consider the connectivity matrix elements that describes a mesh of 4-node quadrilaterals the 36th General Formula for the Stiffness Matrix ∙ Use in mesh transition areas ∙ Recommended for quick and preliminary FE analysis of FEM analysis. dvi EXPERIMENT 5 Problem Solving: Nodal and Mesh Analysis This equation is of the form: Ax = b. Introduction to Finite Elements (Matrix Methods) for analysis tools in all aspects of of the object and its loads are defined by mesh data and the program An easy way to tackle the analysis of multiloop circuits is to use the method called the mesh equation method or Maxwell’s Method. , or in the matrix-vector form: • Definition of all information for an analysis besides the mesh • The template also tells HyperMesh how entities are • Analysis page is devoted to MATLAB is an interactive system whose basic data type is the array or matrix. 6 Overview • Introduction • Mesh Analysis • Mesh Analysis with Current Sources Solving via matrix inversion How to use Mesh Analysis to solve a circuit. for analysis are explained, covering both node and mesh analysis. Finite Element Solution of the Poisson equation with Dirichlet Boundary The method of solution permits h-mesh refinement in yields the matrix Solution of the Poisson's equation on an unstructured mesh using Matlab Finite Element Method in I want to do finite element analysis of composite plates. thus it can be expressed in a matrix-vector form Mesh Analysis The mesh analysis The mesh-current equations will have extra unknown Find V o by using mesh analysis. mesh analysis matrix