A novel finite element scheme

in the analysis of thermal barrier coating components

Dubravka Mijuca

University UNION Cara Dusana, 62-64

11000 Belgrade, Serbia

A new fully three-dimensional reliable multifield primal-mixed finite element approach for the thermo-mechanical analysis of thermal barrier coating components, up to micron sized thickness of the coating, is examined. The essential contribution is that all variables of interest, temperature and heat flux, and displacements and stresses, are solution variables, simultaneously calculated from the semi-coupled systems of primal-mixed finite element equations, respectively. In addition, as a novelty, continuity of the trial and test stress shape functions is enforced, although it represents local violation of discontinuity of some stress components on the material interfaces. It will be shown that such restriction do not deteriorate results. Besides, it is more natural approach since heat flux and stress fields are continuous over each of the material subdomain of the coated body. The proposed hexahedral finite element HC8/27 passes low and high order convergence and efficiency tests in transient heat transfer and elastostatic analyses. That is, it is solvable, stable, accurate and efficient, shortly it is reliable. Consequently, these properties made him perfect candidate for the use in the analysis of thermal barrier coating components. It is unlike to classical displacement finite element approach which suffers from the finite element aspect ratio restriction and will exhibits locking behavior in present applications. In order to minimize the accuracy error and enable introduction of the stress constraints, tensorial character of the present finite element equations is fully respected. Several numerical examples of temperature and mechanical and thermal stress analysis of the coated body are examined to enlighten the efficiency of the present approach.

Increased thermal efficiency and integrity of materials in high-temperature environments are essential requirements in engineering structures. Therefore, either alone, or as an integral part of thermo-mechanical analysis, the heat transfer analysis is of great importance in the analysis and design of engineering components. Particularly, the demand on aero-engine performance and reliability increases as well. There is a need for aero-engines with a large thrust and a high ratio of thrust to weigh. Increasing temperatures in the front of a turbine is one of the most efficient ways to increase thrust and ratio of thrust to weigh. When the temperature before a turbine is higher than 1300K, a blade made of single metal can not undergo such a high temperature. Therefore, the use of thermal barrier coating (TBC) technology is needed which can decrease the peak temperature on substance by 100 - 170њC.

Namely, some base metals will fail because their high temperature limit is lower than the service temperature. Therefore, coating system is applied to the surfaces in contact with the heat. The coating acts as a shield dissipating the heat before it reaches the base metal. Typical applications can be also: jet engine parts, fuel injectors, burner cans, exhaust flaps.

However, whenever thermal barrier coating peels off, the substance behind will lost its structural integrity, that is, over-heat and burn out. These failures are mainly due to the distinctions between the thermal expansion coefficients, that is, physical properties of the coating and substrate materials. Therefore, additional thermal stresses can arise in the coatings, initiating cracks in the coating or resulting in drop off of its fragments from the blade surfaces. As a result, with further running, there is a risk of coating crack propagation into the substrate material, causing its failure on the whole.

Since analytical solutions are unavailable in these situations, the commonly used thermal stress analysis tools of coated components are engineering simplified method, finite element method and boundary element method.

The engineering simplified method (ESM) is easy to use, but its accuracy and effectiveness depend on the stiffness ratio of the coatings to the substrate. In addition, unrealistic constant stress distribution in the coating is assumed along the thickness direction. On the other hand, the primal finite element (PFEM) schemes are abundantly used for over several decades and implemented in the number of the commercial codes. Unfortunately the use of the geometrically non-reduced finite elements, as hexahedral finite element families based on the popular primal finite element approaches, which allow us to have an insight in behavior of structure in all dimensions, is impossible in the present applications because of finite element aspect ratio restriction. Namely, it is due the intrinsic properties of the primal hexahedral finite element which will lock if the aspect ratio of its maximal axial dimensions is lesser then approximately 70 percent, especially in mechanical analysis. The usual engineering approach in these situations is to use very fine finite element mesh, where size of the finite element is of the size of the coating thickness. The obvious consequence will be the huge demand of the storage space and CPU time for the computation. However, less obvious consequence is the loss of the simulation accuracy and accumulation of the errors, if the model problem is not well posed.

Another finite element approach is the mixed finite element (MFEM) approach, which have more than one solution variable in the resulting formulation. If all fields of variables are of the same dimensionality the resulting multifield approach is called mixed, otherwise it is called hybrid. If it is constructed over the stable finite element configurations, it will overcome abovementioned locking problems and to increase the accuracy of dual variable field. Let's remember that the dual variables (e.g. stress) is usually of greater importance than primal variable. The mixed methods are frequently used in fluid mechanics. Theirs efficient and reliable use in solid mechanics is recently demonstrated by the present author. It will be shown, that present hexahedral primal-mixed finite element approach is reliable and robust in the analysis of thermal coating barriers, where coating thickness is of micron size, also.

Another numerical method used to analyze stresses of coating system is conventional boundary element method (BEM), which encounters problems if the domain of the problem is not convex. Similarly to FEM, very fine boundary elements, with a size of the coating thickness, should be used. Nowadays, thermal stress analysis of coating systems by the BEM for thin body stress analysis was limited to problems with a constant temperature field.

Consequently, a primal-mixed method for treating linear coupled thermoelastic problems and the quasi-static or transient case, is presently used in semi-coupled procedure to calculate thermomechanical behavior of coated bodies under thermal and mechanical loads. The approach is fully three-dimensional per geometry, per solution variable space, and per governing and constitutive equations. The finite element used is hexahedral hierarchical HC8/27 in both type of field analysis. It will be shown that there is no consistency error between calculated mechanical strain field and thermal strain field, which makes present approach reliable from the point of view of multifield analysis, that is, semi-coupling of thermal and mechanical fields, also. The thermal analysis capabilities of the present approach include conduction, convection, and radiation analyses, and the structural analysis presently include static analysis. It should be noted that present model problem will have drastic distortions of the finite elements in the mesh, where aspect ratio of finite element maximal axial dimensions will be up to remarkable 6´10⁵. It is proven that present approach is reliable and efficient. Nevertheless, the present analysis illustrated through some numerical tests, show that a direct approximation for the stress eliminates the generation of spurious displacement and stress modes, although this is not a priori guaranteed for a general mixed finite element schemes. Namely, it is proven that present finite element scheme do not exhibit stress oscillation near the stress singular points. In addition, although present approach has greater number of unknowns than primal one, it is proven in that for the prescribed accuracy the execution time is even smaller. In addition, the computational cost it is governed also by the quality of the method used for the solution of the large systems of equations.

The fully three-dimensional primal-mixed finite element approach in thermo-mechanical analysis of thermal barrier coated components is examined in detail. All solution variables: temperature and heat flux, and displacement and stress, are simultaneously calculated from semi-coupled system of thermal and mechanical field equations. As a novelty, the interelement continuity of heat flux and stress shape functions is intentionally enforced. It is shown that it does not deteriorate target results in the region of material interface. The mathematical aspects of convergence of the proposed finite element configurations HC8/27, is carefully analyzed also. The success of the present fully three-dimensional approach is due to its robustness and reliability. Further, obtained solutions converge to the target result, regardless of the complexity of the geometry, or presence of the abrupt material changes, non-smoothness of boundary conditions per dual variables or distortion of the finite element mesh. Nevertheless, one of the main potentials of the present approach is in overcoming of the well-known transition problem of connecting finite elements of different types and dimensions. Consequently, present finite elements are robust in micron-sized coated components analysis. In addition, in connection with specially designed HSL solution routine MA47+MC30, the present finite element approach is time efficient, measured as the accuracy versus the solution time. Finally, the present approach can be easily implemented in any existing pre and postprocessing environment developed for primal finite element approach, as it is presently implemented in Straus7 software package.