Mass matrix in finite element software

To this end, integrands of mass matrix entries get separated to two. The present version z88aurora v4 offers, in addition to static strength analysis modules such as nonlinear strength calculations large displacements, simulations with nonlinear materials, natural frequency, static thermal analysis and a contact module. Depending on the modelling software you use for your finite element analysis fea, your modelling process often requires an understanding of consistent units coherent units. Shape functions and node numbering related to 3d reference elements are shown in the following table.

The finite element method fem is a numerical approximation method. The primary characteristics of a finite element are embodied in the element stiffness matrix. How to construct mass matrix and stiffness matrix equations. This is done fully automatic and works for any time derivative. New concepts for finiteelement mass matrix formulations. Here, the functions x1t and x2t represent state variables of the system.

For discrete approximations of continuum mechanics as in the finite element method, there may be more than one way to construct the mass matrix, depending on desired computational and accuracy performance. Mass matrix construction the master mass matrix is built. Constitutive model and computational details a structural analysis method for the longterm response of wood structures is presented in this report. The finite element method fem is the most widely used method for solving problems of engineering and mathematical models. Mass matrices for individual elements are formed in local coordinates, transformed to global, and merged into the master mass matrix following exactly the same techniques used for k. Correlation of a cantilever beam using beam theory, finite.

Although in implicit time integration, we will sometimes use a lump mass matrix, sometimes a banded mass matrix, a consistent mass matrix. Examples and problems comparison of different elements behaviour under bending. You can clarify what exactly you need, but basically the stiffness matrix comes from strain energy stress times strain and the mass matrix comes from kinetic energy mass times velocity squared. Dynamic analysis in finite element methods consistent mass matrix lumped mass matrix problems. This is balanced by some negative aspects that are examined in some detail later. Can you please point me out to the resources on the finite element method, where the procedure of assembling a mass matrix is described in detail. Vibration analysis using finite element method part 1dynamic analysis. You need to start from the formulation of the governing equations.

Typical problem areas of interest include the traditional fields of structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential. The fem is a particular numerical method for solving. The finite element method is exactly this type of method a numerical method for the solution of pdes. Solve eigenvalue problem for a number of eigenmodes. Solved many unsolved problems on bar, beam, heat transfer and trusses etc. For this reason, i1 and i2 must not be zero for the cbeam element. Create the following system of differential algebraic equations. Rather than imposing the constraint relations on the global stiffness or mass matrix as the conventional approach, this procedure is based on the element formulation level in that the element matrices and vectors are properly converted to account for the effect of constraint relations.

As a general rule, the construction of the master mass matrix m largely parallels of the master stiffness matrix k. List of finite element software packages wikipedia. The advantage of having diagonal mass matrix in an eigenvalue analysis is the ease and reduction of computational effort. Ansys structural analysis software enables you to solve complex structural engineering problems and make better, faster design decisions. It is a method of investigating the behavior of complex structures by breaking them down into smaller, simpler pieces. Assemble finite element matrices matlab assemblefematrices. Herein, we present an alternative semianalytical approach for mass matrix evaluation, resulting in an accurate, efficient and easytoimplement integration rule. Extract mass matrix and right side of semilinear system of. The timedomain transient method in structural finite element.

Convert a semilinear system of differential algebraic equations to mass matrix form. The matrix is usually referred to as the stiffness matrix, while the matrix is dubbe d th e mass matr ix. Perform further analysis time history or response spectra. Model attributes, such as coefficients, material properties, boundary conditions, and so. They do, however, get more development for their dollar. Effect of mass matrix formulation schemes on dynamics of structures. Fma, thus mass must be sluginch this mass weighs 386lbf on earth. Currently, numerical integration schemes, mostly based on gauss quadrature, are widely used. Model attributes, such as coefficients, material properties, boundary conditions, and so on, must not depend on time or solution. On semianalytical integration specified for mass matrix of finite. Finite element method finite element analysis youtube. Nonlinear transient dynamic analysis by explicit finite element with iterative consistent mass matrix communications in numerical methods in engineering, vol. New easytoimplement, efficient, formulas specified for widelyused eightnode brick element. The formulation for the cbeam element is based on a flexibility approach.

Dynamic analysis in finite element methods consistent mass matrix. Lumped mass matrix in explicit finite element method for transient. Theory, implementation, and practice november 9, 2010 springer. Approximating a consistent mass matrix m 1, m 2 with a lumped mass matrix e m 1, e m 2 will reduce the accuracy of the finite element method, however it is shown in 39 that the lumped. A linearized finite element model can be summarized in terms of matrices. Although the assumption that damping is proportional to the mass and stiffness matrix has no sufficient physical basis, we know very little about the distribution of damping, and practice has proven that rayleigh damping is effective in the finite element analysis thus is widely used. Typical structural elements are mass, truss, beam, membrane, plane stressplane strain, thin plate, thin shell, thick plate, 3 dimensional solid with a variety of shape functions. When solving time dependent pdes using the finite element method, for example say the heat equation, if we use explicit time stepping then we have to solve a linear system because of the mass matrix.

In finite element analysis of two and three dimensional problems the isoparametric concept is particularly useful shape functions are defined on the parent or master element each element on the mesh has exactly the same shape functions shape functions are used for interpolating the element coordinates and deflections parent element. The fem consists in discretizing a continuum into small. The lumped mass matrix is formed by lumping of mass at the nodes of the element. Finite element analysis a typical finite element analysis on a software system requires the following information.

The advantage of having diagonal mass matrix in an eigenvalue analysis is the ease and reduction of computational effort as well as requirement of lesser storage space. This is obtained by using the shape functions 2 for the elements and is given by 3 this involves off diagonal entries and also referred in the cae community as full or coupled mass matrix in fea softwares. Introduction to dynamics using finite elements analysis fea. Lets try with using only one integration point for integrating the element shear stiffness matrix element shear stiffness matrix of an element with length l e and one integration points stiffness matrix of the timoshenko beam 2. These algorithms are designed to exploit the sparsity of matrices that depend. With the finite element analysis fea solvers available in the suite, you can customize and automate solutions for your structural mechanics problems and parameterize them to analyze multiple design scenarios. If the size of the standardform matrix is relatively small, eigenvalues and eigenvectors of the matrix can be directly solved within a relatively short period of time. Typical problem areas of interest include the traditional fields of structural analysis, heat transfer, fluid flow, mass transport. The purpose of finite element analysis fea software is to reduce the number of prototypes and experiments that have to be run when designing, optimizing, or controlling a device or process. For a structural finite element, the stiffness matrix contains the geometric and material behavior information that indicates the resistance of the element to deformation when. The difference in the mass center of gravity and the shear center is significant.

With an explicit time integration, we generally use only the lump mass matrix. Jul 12, 2017 you need to start from the formulation of the governing equations. Find such that, where is the stiffness matrix and is the mass matrix. Fem assemblefematricesmodel returns a structural array containing finite element matrices. The mass matrix and the stiffness matrix in finite element. Effect of mass matrix formulation schemes on dynamics of. Volumetric finite elements description robot structural. Consistent engineering units in finite element analysis. Similar to the thermal energy conservation referenced above, it is possible to derive the equations for the conservation of momentum and mass that form the basis for fluid dynamics.

In a matrix generation analysis, you can output the stiffness, mass, viscous damping. The matrix is usually referred to as the stiffness matrix, while the matrix is dubbed the mass matrix. Does abaqus software calculate stiffness matrix and mass matrix. On semianalytical integration specified for mass matrix of. For example, a lumpedmass method, in which the deformation of each element is ignored, creates a diagonal mass matrix and negates the need to integrate mass across the deformed element. To compare the different elements described earlier, the simply supported beam with the distributed load shown in figure 1 was modelled in the finite element analysis software abaqus with various different element types. In the finite element derivation of 3d free vibration problems, the mass and stiffness matrices can be transformed into a single standardform matrix.

The use of the lumped mass matrix is one of the important features making the explicit finite element method efficient and practical 2, 3. Interpretation of stiffness matrix and mass matrix in finite. It is a diagonal matrix obtained by either row or column. Now, the first step in any finite element analysis must, of course, be the step of idealizing the total. The method has been developed for the finite element calculation software abaqus applying a user defined material model. Dynamic analysis in finite element methods consistent mass. A method to model wood by using abaqus finite element software. Introduction to finite element analysis fea or finite. Finite element modeling for dynamic analysis of large and complex structures such as ship hulls, offshore structures, aerospace structures etc, introduces a very. In dynamic analysis using explicit time integration, we have the mass matrix, the same mass matrix, generally, that we have here. An efficient matrix tridiagonalization method for 3d finite.

The second order time derivative corresponds to the mass matrix and a first order time derivative corresponds to the damping matrix. Alternatively, using diagonal or lumped mass formulation not only reduces the computational cost but can also simplify the program coding. The mass matrix is usually not updated in lagrangian fem because mass is. Detailed explanation of the finite element method fem. A method to model wood by using abaqus finite element software part 1. Composite damping can define a critical damping ratio. The system also contains symbolic parameters r and m, and the function ft, x1. By closer inspection, it turns out that the row sum of the matrix elements in m is 1 one, people just lump together the matrix elements of m on the diagonal, which then becomes an identity matrix. Volumetric structures in robot are modeled using isoparametric volumetric finite elements with displacement field approximation based on the 1st order shape functions. The elements are connected to each other at the nodes. Describe the terms lumped mass matrix and consistent mass matrix and identify which formulation is appropriate to elements being used.

The finite element method fem is an engineering tool that allows solving several types of engineering problems. A computer program for automatic generation of stiffness and mass. Constraint relation implementation for finite element. What is the physical significance of the stiffness matrix in. Demonstration of accuracy advantage based on coarse mesh generated with commercial software. Many of the matrix elements are zero, so that reduces to references. Assuming that you are doing static analysis, this would generally be a representation of newtons second law. Mass matrix construction the master mass matrix is built up from element contributions, and we start at that level. Finite element analysis fea software comsol multiphysics. General fo rm of t he fini t e ele ment method edit in gener a l, the fi n ite e lement method is characterized by the following process. This does not necessarily mean that companies and research institutes save money by adopting fea.

Furthermore, the lumped mass matrix has shown to be. Spatial numerical integration is essential for finite element analysis. A method to model wood by using abaqus finite element. But avoid asking for help, clarification, or responding to other answers.

The bar in actuality, is supported here, but as i mentioned earlier, we remove that support in our finite element formulation, and introduce, in fact, a displacement degree of freedom there. There are two strategies in the finite element analysis of dynamic problems related to natural frequency determination viz. Lumped mass matrix in explicit finite element method for. In finiteelement programming, the computation of this matrix is usually performed elementwise, looping over all the elements and adding the nonzero contributions to the global matrix. Now, the reason that there is a mass matrix field in the data structure of the discretized pde is to, for example, model rayleigh damping. Mass matrix computation in the finite element method.

These smaller pieces of structure are called elements. Illustrate the approximate nature of finite element analysis, through dynamic examples chosen from your industry sector. A procedure for implementing constraint relations among finite element nodal degrees of freedom is outlined. A computer program to generate automatically the stiffness and mass matrices in finite element analysis is introduced. What is lumped mass modification of finite element method. If the answer is yes, then the software must exclude the elements reached yield. These longitudinal deformation are called geometric deformations. For a structural finite element, the stiffness matrix contains the geometric and material behavior information that indicates the resistance of the element to deformation when subjected to loading. Assembly from the elemental to global mass matrix is the same as the stiffness matrix. Thanks for contributing an answer to physics stack exchange. How is the mass matrix formed in finite element methods.