1. The study of the mechanical behavior of structures#

1.1. A general code#

code_aster is a general purpose code for studying the mechanical behavior of structures.

The priority field of application is that of the mechanics of deformable solids: this justifies the number of functionalities associated with the mechanical phenomenon. However, the study of the mechanical behavior of industrial components first requires modeling the stresses to which they are subjected, or physical phenomena that modify the parameters of this behavior (internal or external fluid, temperature, change in metallurgical phases, changes in metallurgical phases, forces of electro-magnetic origin, etc.). For these reasons, code_aster offers several possibilities for « chaining » the mechanical phenomenon with thermal or acoustic phenomena, or with other software, as well as a « kit » for constructing coupled thermo-hydro-mechanical problems.

Although code_aster can be used for many structural calculation problems (general code), it was developed in particular to allow the study of the components of equipment or machines used in the field of electricity production and transport. Thus priority was given to the modeling of metal structures, geomaterials and reinforced concrete structure components. Very few models available for composite materials or elastomers will be found.

Nonlinear analyses, both in mechanics and in thermal, are at the heart of code_aster: their efficient processing required the development of efficient and relatively easy to use algorithms, even if the aim is not to make them work in a « black box ». For complex studies, it is therefore necessary to understand the nature of the operations carried out by the code, in order to be able to pilot them optimally: we then refer to the theoretical instructions giving the details of the models and methods, grouped together in the Reference Manual.

1.2. Methodology for calculating with code_aster#

A structure calculation carried out with code_aster consists of the sequence of a certain number of commands described in a « command file » in text format. The engine and interpreter for this command file is the PYTHON script language. It is therefore possible to use all the functionalities provided by PYTHON. See in particular the documentation [U1.03.01], [U1.03.02] and the examples of use [U1.05.00] and [U1.05.01]. Each command (for example, mesh reading, material data assignment, linear static calculation) produces a « result concept, » which defines a data structure that the user can manipulate and reuse in subsequent calculation commands.

The syntax of all commands is described and commented on in the U4 and U7 manuals in the User documentation.

In order to simplify the user’s task, there are global commands that group together the appropriate sequence of operations for a certain number of calculation cases (for example static linear - command MECA_STATIQUE, static non-linear - command STAT_NON_LINE, thermal non-linear - command, non-linear thermal - command THER_NON_LINE, etc.). Some have been developed directly in an integrated manner, others are macro commands in Python that manage calls to various unit commands (like ASSEMBLAGE which allows you to calculate and assemble the mass, damping and stiffness matrices of a structure).

There are also macro commands specially dedicated to certain applications.

At the end of a calculation, it is often possible to enrich the computer object containing the « result concept » obtained, by carrying out other calculations a posteriori: for example, from the displacement field and the constraints at the Gauss points obtained in a mechanical calculation, we can calculate the deformation field, the stress field interpolated at the nodes, etc.

These post-processing operations are carried out by operators CALC_CHAMP, POST_CHAMP,,, POST_ELEM, MACR_LIGNE_COUPE, POST_RELEVE_T, etc.

1.3. Phenomena, models, finite elements and behaviors#

1.3.1. Notions#

In code_aster, PHENOMENE (command AFFE_MODELE) is a family of physical problems that use similar unknowns and conservation equations: for example, the mechanical phenomenon uses displacement unknowns, the thermal phenomenon uses temperature unknowns. Depending on the modeling used, the number of unknowns of this type may vary (for example, we only need one 3D temperature unknown for each node, but 3 unknowns are used for shells).

In code_aster, MODELISATION (command AFFE_MODELE) describes how the continuous equations governing a given phenomenon are discretized, with the help of possible additional hypotheses (plane deformations, beam model, shell model…). In mechanics, for example, we can find 3D models, plane deformations, plane stresses, shells, plates, Euler beams, Timoshenko beams, pipes, etc… Each of the models uses a specific set of degrees of freedom: for example, displacements in the three directions of space for 3D continuous medium models, three displacements and three rotations for shells, etc.

The PHENOMENE/MODELISATION pair makes it possible to assign bijectively a type of finite element to each type of geometric mesh.

In code_aster, we call the « finite element », for a given PHENOMENE/MODELISATION couple, the set consisting of:

  • The geometric nature of the support cell (representing a piece of volume or border): hexahedron, tetrahedron, triangle, quadrangle, segment… );

  • The choice of interpolations for geometry and for unknowns (shape functions);

  • The calculation « options » that the element can calculate (the operations for which the calculation of the appropriate integrals has been programmed: for example, elementary stiffness term, elementary surface force term, elementary mass term, etc.).

  • Numerical integration schemes (Gaussian quadrature formulas for example).

Behaviour is basically a physical concept linked to the properties of the material. It is then expressed mathematically. For example, in mechanics, the behavioral relationship is the relationship that links the stress field to the deformation field. During a calculation, the behavioral relationship is calculated at each integration point (or Gauss point).

code_aster distinguishes between the characteristics of the material: commands DEFI_MATERIAU and AFFE_MATERIAU and the behavioral relationship (keyword COMPORTEMENT).

1.3.2. The phenomena available in code_aster#

code_aster is a software that mainly deals with solid and structural mechanics but to complete the representation of the operating environment of mechanical components, the choice was made to include in code_aster functionalities allowing the modeling of phenomena frequently associated with mechanical phenomena such as thermal or acoustic.

1.3.2.1. The mechanical phenomenon#

The mechanical phenomenon is modelled to achieve two main objectives:

Firstly:

The determination of the internal state, in particular the stress state at each point of a structure, under various stresses representing the operating conditions. Knowledge of this state of stress makes it possible to continue the analysis of mechanical behavior:

  • to verify compliance with the construction rules specific to each type of structure, in particular the Design or Construction Rules (RCC… );

  • to analyze the harmfulness of defects and their possible propagation: defects inherent in the process of developing the component or structure (inclusions, geometric imperfections, etc.) or resulting from operating conditions (cracking, erosion, etc.) or resulting from operating conditions (cracking, erosion, etc.) );

  • for the study of behavior under cyclic loading and fatigue analysis;

  • for the prediction of allowable loads with changes in the internal state.

Secondly:

The determination of the deformed configuration induced by a permanent (static) loading or resulting from a slow (almost static) or faster (dynamic) evolution of loads or boundary conditions. Knowledge of this deformed configuration, and possibly of the corresponding speeds and accelerations, makes it possible to continue the analysis of mechanical behavior:

  • for vibratory or acoustic behavior;

  • for the transmission of stresses to other structures or components;

  • for the analysis of interactions with neighboring structures to determine operational anomalies or wear parameters that may result.

The levels of modeling that are used to study this phenomenon are:

The nonlinearities taken into account in the mechanical phenomenon are behavioral nonlinearities, geometric nonlinearities (large displacements and large rotations, large deformations, buckling) and contact/friction nonlinearities.

1.3.2.2. The thermal phenomenon#

It makes it possible to determine the thermal response of solid media in a steady state (stationary problem) or transient (evolutionary problem). We can model solid conduction, convective exchange, heat exchange between walls, and radiation to infinity. The thermal phenomenon may include modeling by heating or cooling the metallurgical phase change of steels, which makes it possible to simulate heat treatment or welding operations (the identification of the behavior is based on experimental diagrams TRC).

By analogy of the equations solved, the thermal phenomenon can also be used to model hydration (the unknown is the degree of hydration) or the drying of concrete (the unknown is the water concentration).

1.3.2.3. The acoustic phenomenon#

The acoustic phenomenon is modelled to achieve two main objectives:

  • The study of acoustic propagation in a closed environment corresponding to the Helmholtz equation in a compressible fluid, for propagation domains with complex topology. Knowledge of the pressure field makes it possible to continue the acoustic analysis to determine the field of sound levels (expressed in \(\mathit{dB}\)) and the fields of active and reactive acoustic intensity.

  • The study of coupled 3D vibro-acoustic problems corresponding to the vibratory behavior of a structure in a limited domain of compressible, non-viscous fluid.

1.3.3. Coupled phenomena available in code_aster#

So that there is no ambiguity, a distinction will be made between:

  • The linking of two phenomena (or weak coupling): preliminary study of the first phenomenon whose results are used as data for the second;

  • The complete coupling (or strong coupling) of several phenomena: simultaneous resolution of the phenomena with effectively coupled equations.

1.3.3.1. The internal chains of code_aster#

Chaining can be done within code_aster or between this one and external software.

The chains currently carried out within code_aster are as follows:

  • Thermal — mechanical: all the mechanical characteristics of materials can depend on temperature and the algorithms available for the mechanical phenomenon make it possible to exploit the results of a prior thermal calculation (anelastic deformations: thermal expansion, concrete shrinkage, etc.) carried out on a possibly different mesh;

  • Thermal — metallurgical: after a thermal calculation, it consists in calculating the proportions of the various metallurgical phases of steels;

  • Thermal — metallurgical — mechanical: taking into account four mechanical effects of metallurgical transformations (phase change deformation, modification of mechanical characteristics, transformation plasticity, transformation plasticity, work-hardening restoration);

  • Electrical — mechanical: electrical coupling is limited to taking into account the Laplace forces induced by short-circuit currents in electrical cables;

  • Fluid — mechanics: assignment of the pressure field on a wall deduced from a fluid mechanics calculation.

1.3.3.2. Thermo-hydro-mechanical couplings#

Saturated or unsaturated porous media (geomaterials, soils, concrete) must be studied by combining the three mechanical, thermal and hydraulic equations. The user chooses the behaviors he wants to use from a kit of thermo-hydro-mechanical models called THM. He can thus choose whether or not to take into account the effect of temperature, and to represent one or two pressures. The choice of each of the behaviors associated with the phenomena selected is also made within this framework.

1.3.3.3. Couplings for fluid-structure interaction#

Three types of couplings are available in the field of fluid-structure interaction:

  • The calculation of the natural modes of a structure containing (or immersed in) a fluid at rest (with or without a free surface);

  • The calculation of the vibrations of a structure in a flow and the estimation of the damage resulting from vibration fatigue or wear;

  • Taking into account a boundary condition such as an infinite fluid domain.

1.4. Analysis methods#

To implement the various models, several analysis methods are available that correspond to different processes of application of stresses.

1.4.1. Static mechanics#

Static analysis: it corresponds to permanent stresses for the treatment of stationary heat and thermo-mechanics. For linear analyses, the results obtained can be combined linearly, as required, and can be used to describe the initial state of an evolutionary process.

Quasi-static analysis: for all mechanical processes where inertia phenomena can be overlooked, implicit incremental algorithms are available to solve non-linear behavior equations taking into account evolutionary loads and boundary conditions.

1.4.2. Thermal#

Stationary analysis: in linear and non-linear thermics, calculation of the thermal equilibrium state.

Transient analysis: in linear and non-linear thermics, with possible consideration of metallurgical effects for metals and hydration and drying for concrete, as well as for thermo-hydro-mechanical problems by neglecting the effects of inertia on the mechanical part.

1.4.3. Dynamic mechanics#

For processes where the effects of inertia and propagation must be taken into consideration, one speaks of dynamic analysis.

Physical analysis is the resolution of equations in the classical basis of physical degrees of freedom.

Modal-based analysis consists of a preliminary calculation of the eigenvalues and vectors of the structure, representing the « vibratory » state of the system. The equations to be solved are then projected onto a basis of eigenvectors: the number of degrees of freedom of the system to be solved is proportional to the size of the modal base used.

For these two types of analyses based on a physical or modal basis, the response calculation can be performed in time or in harmony (in the linear case).

For seismic analysis, it is also possible to formulate the problem in imposed motion in a relative frame of reference (without the driving movement).

Linear dynamic analyses can be done by including the effects, of the second order on stiffness, of the initial static stresses calculated beforehand (geometric rigidity, centrifugal stiffness).

For nonlinear problems, two analysis methods are available:

  • Analysis by modal recombination with localized nonlinear boundary conditions for shock problems (operator DYNA_VIBRA);

  • Nonlinear dynamic analysis on a physical basis (operator DYNA_NON_LINE).

1.4.4. Sub-structuring#

Sub-structuring consists in grouping several finite elements within a macro-element and in « condensing » all of their rigidity onto the (fewer) degrees of freedom of this macro-element.

The resolution of the global problem is then limited to the determination of the unknowns carried by the macro-elements and then to the calculation of the unknowns carried by each « small » element independently within each of the macro-elements.

The advantages of this method are the savings in time and memory, when the complete structure is formed by elements reproduced several times by translation or rotation.

In dynamics, modal analysis and harmonic or transient response calculation can be performed using classical dynamic substructuring by Craig-Bampton, Mac Neal methods or for the so-called interface mode method.

For structures with cyclic symmetry, the available methods make it possible to calculate the natural modes of the global structure from the dynamic behavior of a base sector.