This allows to express the mechanical problem at hand in terms of the stationary condition of the Total Potential Energy (TPE). the stress solution is found as a derivative of some potential energy function. The mechanical formulation is based on the kinematics of finite deformation and, for the description of the material behaviour, on isotropic hyperelasticity, i.e. The applicative context is the nonlinear analysis of solids and structures, see, by showing the formulation and the subsequent MATLAB® coding of some typical structural and solid finite elements.
MATLAB SYMBOLIC TOOLBOX GRADIENT MATRIX CODE
This result is obtained by condensing the development process going from the mathematical formulation to the prototype code implementation in a few lines of MATLAB® symbolic instructions.
MATLAB SYMBOLIC TOOLBOX GRADIENT MATRIX HOW TO
Its intent is to show how to use MATLAB®, in particular by exploiting the capabilities of the Symbolic Math Toolbox™, to produce numerical approximations of a given solid mechanics problem in a way that the usual gap between the theoretical formulation and its actual implementation in a code is not perceived. The present work, quite far from being an alternative to the hugely developed and rich packages previously cited, proposes a basic approach for the quick early phase developing of solid mechanics finite elements formulation. Inside this generic form the specific differential problem to be solved must to be fitted by the user, see for example, but usually with no control over the discretization technique used by the solver. Other packages instead solves a generic system of Partial Differential Equations (PDEs) subjected to boundary and initial conditions. In both cases the user must define the procedures or functions needed in order to assign the desired new finite element to be used within the analysis framework already available in the library. This is the case of open source FEM libraries or commercial packages listed in. Such solutions typically refer to a specific context by keeping fixed the physical problem but letting open the specific instance of discretization technique which, however, is fixed too. However some solutions capable to assist the developer in this process already exist ant it is worth to mention some ones. This makes possible to extend the validation of the conceived numerical model by performing the analysis of larger sized problems.Īs already said, the previously described work-flow is lengthy and it is often characterised by a gap between the theoretical formulation and its implementation in a numerical code. The additional last step can be the production of an executable by using compiled programming languages such as C/C++ or fortran. In any case it is necessary to go back and to repeat the process just described. Often this phase can highlight also flaws in the mathematical model or in the discretization technique. This prototype code allows to perform basic tests with the aim to check the effectiveness of the adopted model with respect to well known situations and to check for the presence of bugs. The discrete model so defined is usually inserted into a prototype code, often by using again an algebraic manipulator but the use of compiled programming languages is also possible if not common. MATLAB® is certainly one of the state-of-the-art mathematical softwares available for performing numeric or symbolic analyses, but it is not the only one and a quite long list, see, of packages offering very similar features is available. This phase is characterised by the evaluation and the analysis of these operators through an algebraic manipulator such as MATLAB®. For example, in the case of the analysis of solid mechanics problems by FEM, important discrete operators are the mechanical response vector of the finite element and its tangent stiffness matrix. This operation leads to the identification of the needed discrete operators which define the computational model. The most popular technique is the Finite Element Method (FEM), see, but now the number of computational approximation techniques is very large and a synthetic summary can be tried only by citing some of these less conventional methods: mixed finite element methods partition of unity-based discontinuous finite elements meshless methods discontinuous Galerkin methods.
The subsequent step is the introduction of a numerical approximation technique. At present time the definition of a mathematical model is at the basis of any serious attempt to obtain previsions in any engineering application, but not only in the engineering field. The typical starting point is the formulation of a mathematical model where the main physical or real world phenomena to be described are established.
The developing of finite element formulations, standard or new ones, requires a lengthy process which involves several steps.
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