{"id":16570,"date":"2017-03-24T14:52:38","date_gmt":"2017-03-24T13:52:38","guid":{"rendered":"\/en\/interdepartmental-laboratory-of-numerical-modeling\/badania\/metoda-elementow-skonczonych\/"},"modified":"2017-04-15T21:08:29","modified_gmt":"2017-04-15T19:08:29","slug":"finite-element-method","status":"publish","type":"page","link":"https:\/\/www.ipan.lublin.pl\/en\/interdepartmental-laboratory-of-numerical-modeling\/research\/finite-element-method\/","title":{"rendered":"Finite Element Method"},"content":{"rendered":"

[vc_row css_animation=”” row_type=”row” use_row_as_full_screen_section=”no” type=”full_width” angled_section=”no” text_align=”left” background_image_as_pattern=”without_pattern”][vc_column][vc_column_text]In the field of agricultural sciences, numerical modelling has proven to be a valuable tool in finding solutions to practical and scientific issues. Such models are necessary to understand the structure related properties of plants, like fruits and vegetables, and bring, in the next step, possibilities of extension by nano-structural features (like cell wall composition). This knowledge will be useful for engineering and improve fruits and vegetables quality.<\/p>\n

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The basic idea behind finite element method (FEM) is subdivision of a spatial domain of the problem into a simpler parts called the finite elements. The solution in global domain is obtained as a result of assembly of local solutions for the finite elements. Finite element method allows studies of systems with large complexity, irregular shapes and nonhomogeneous material properties.<\/p>\n

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Studies carried out by the Interdepartmental Laboratory of Numerical Modeling are aimed to create computational models of plant tissues at different spatial scales.<\/p>\n

Modelling of plant tissues mechanical properties<\/strong><\/p>\n

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Physical properties plant materials are related to several micro- and macroscopic morphological features such as the spatial arrangement and shape of cells, the number of intercellular spaces, turgor, the nano-composition of cell walls and the degree of degradation of the middle lamellae. The experimental analysis of the structure related micromechanical properties of plants\u00a0 has been constrained by the lack of technology available for conducting reliable measurements at such scales. Deeper understanding of mechanical properties of plant tissues can be achieved by means of the numerical models of tissue deformation under various load conditions.<\/p>\n

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Up to now, the majority of models described in the literature was based on the principles of classical continuum mechanics. Most often, tissues were described by empirical or analytical models, as uniformly stressed structures, with symmetry and uniformity assumptions on cell shape and cell wall deformations. Although this approach allowed for fairly accurate predictions of the behavior of the plant materials in engineering applications, it was unable to provide an explanation of the micro-scale mechanisms underlying deformation and failure of tissue.<\/p>\n

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The research was co-founded by National Science Centre of Poland (research grant no. 2011\/01\/N\/NZ9\/02496).<\/strong><\/p>\n

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Tissue structure modelling
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One of the important steps in solving a problem, before the appropriate analysis with FEM, is to create a virtual model of the tested object. The virtual model is defined by its geometry, material properties and boundary conditions such as loads and the type of supports. It is important to create a proper model with the highest possible degree of accuracy regarding real shape reconstruction; but on the other hand, the model must be simple enough to allow for efficient calculations.<\/p>\n

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In the preliminary study three different methods for parameterisation of plant tissues were tested (Pieczywek et al., 2011<\/a>). \u00a0All methods can be applied to images obtained with a confocal scanning laser microscope to create models for the simulation of the mechanical behaviour of biological cellular structures. Vectorisation, Voronoi tessellation and ellipse tessellation were tested. Potato tuber and carrot parenchyma were chosen as examples.[\/vc_column_text][vc_column_text css=”.vc_custom_1490377778912{padding-top: 30px !important;padding-bottom: 30px !important;}”]<\/p>\n

Badania finansowane by\u0142y w ramach projektu nr 2011\/01\/N\/NZ9\/02496\u00a0 Narodowego Centrum Nauki.<\/strong><\/p>\n

[\/vc_column_text][vc_column_text]Modelowanie struktury tkanki<\/strong><\/p>\n

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Jednym z krok\u00f3w poprzedzaj\u0105cych w\u0142a\u015bciw\u0105 analiz\u0119 MES jest utworzenie wirtualnej reprezentacji badanego obiektu. Wirtualny model zdefiniowany jest przez geometri\u0119, w\u0142a\u015bciwo\u015bci materia\u0142owe oraz warunki brzegowe reprezentuj\u0105ce oddzia\u0142ywania zewn\u0119trze oraz wewn\u0119trzne. Istotne jest aby odwzorowa\u0107 modelowany system z mo\u017cliwie najwi\u0119ksz\u0105 dok\u0142adno\u015bci\u0105, przy jednoczesnym zachowaniu pewnych uroszcze\u0144, umo\u017cliwiaj\u0105cych przeprowadzenie oblicze\u0144 w rozs\u0105dnym czasie.<\/p>\n

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W badaniach wst\u0119pnych przetestowane zosta\u0142y trzy r\u00f3\u017cne metody parametryzacji struktury tkanki ro\u015blinnej (Pieczywek et al., 2011<\/a><\/span>). Wszystkie metody zastosowa\u0107 mo\u017cna do obraz\u00f3w mikroskopowych uzyskanych przy u\u017cyciu laserowego mikroskopu konfokalnego (CLSM), aby nast\u0119pnie utworzy\u0107 na ich podstawie wirtualne modele numeryczne struktur kom\u00f3rkowych. Testom poddano metod\u0119 wektoryzacji, teselacji eliptycznej, oraz teselacji Woronoja. Jako materia\u0142 badawczy wybrane zosta\u0142y tkanki parenchymatyczne ziemniaka oraz marchwi.[\/vc_column_text][vc_column_text css=”.vc_custom_1490377922992{padding-top: 30px !important;padding-bottom: 30px !important;}”]\"\"[\/vc_column_text][vc_column_text]Fig. 1.<\/strong> Vectorisation procedure: detection of a single cell boundary, detection of junction points,<\/p>\n

connection of vertices and formation of a virtual cell.[\/vc_column_text][vc_column_text css=”.vc_custom_1490377991776{padding-top: 30px !important;padding-bottom: 30px !important;}”]\"\"[\/vc_column_text][vc_column_text]Fig. 2.<\/strong> The process of creating a Voronoi diagram.[\/vc_column_text][vc_column_text css=”.vc_custom_1490378049909{padding-top: 30px !important;padding-bottom: 30px !important;}”]\"\"[\/vc_column_text][vc_column_text]Fig. 3.<\/strong> The ellipse tessellation algorithm: fitting of ellipses, determination of the intersection points,<\/p>\n

construction of \u201ccutting lines\u201d, the final model.[\/vc_column_text][vc_column_text css=”.vc_custom_1492283117224{padding-top: 30px !important;}”]For each method tested, five geometrical parameters were analysed: area, perimeter, orientation, elongation and a local indicator of spatial association of all individual regions which represented cells. The reconstruction accuracy of the original tissue microstructure by each parameterisation method was investigated by the comparison of the geometrical properties of the cells from the segmentation with their virtual equivalents.[\/vc_column_text][vc_column_text css=”.vc_custom_1490378150302{padding-top: 30px !important;padding-bottom: 30px !important;}”]\"\"[\/vc_column_text][vc_column_text]Fig. 4.<\/strong> Example of the parametrisation methods of the skeleton obtained from confocal scanning microscopy, a) original image of potato tissue, b) vectorisation, c) Voronoi tessellation, d) ellipse tessellation of the same structure.[\/vc_column_text][vc_column_text css=”.vc_custom_1490378185471{padding-top: 30px !important;padding-bottom: 30px !important;}”]\"\"[\/vc_column_text][vc_column_text]Fig. 5.<\/strong> Performance of the parametrisation methods in reconstruction of irregularstructures: a) original image of carrot tissue, b) vectorisation, c) Voronoi tessellation, d) ellipse tessellation of the same structure.[\/vc_column_text][vc_column_text css=”.vc_custom_1492283159147{padding-top: 30px !important;}”]Based on the results, Voronoi tessellation was considered to be inaccurate for tissue modelling. The vectorisation procedure only allowed for reproduction of the general shapes of cells, and the curvature of cell walls was neglected in this method. For both the Voronoi tessellation as well as vectorisation, created cells completely filled the space with no additional gaps and possessed sharp, angular shapes. The best overall reconstruction accuracy was obtained with ellipse tessellation. Models created with this method can be considered as representative equivalents of real tissues in terms of cell area, orientation, perimeter, shape and spatial arrangement.[\/vc_column_text][vc_column_text css=”.vc_custom_1492283170845{padding-top: 30px !important;}”]Micro-scale model of onion epidermis tissue<\/strong><\/p>\n

In the present study a computational model of plant tissue that incorporates micro-scale geometrical features was developed to provide qualitative and quantitative predictions of the mechanical properties of onion (Allium cepa<\/em>) epidermis (Pieczywek et al., 2014<\/a>). Onion epidermal tissue has been chosen as a simple and well-defined system for model validation. The simulations of cellular structure behaviour under various mechanical load conditions were carried out using the finite element method (FEM).\u00a0\u00a0 The finite element method was chosen due to its computational efficiency, flexibility and ability to incorporate geometric nonlinearities. The models were validated against experimental data from a tensile test of the real tissue strips.[\/vc_column_text][vc_column_text css=”.vc_custom_1490378340242{padding-top: 30px !important;padding-bottom: 30px !important;}”]\"\"[\/vc_column_text][vc_column_text]Fig. 6.<\/strong> Video record of tensile test using the real onion epidermis tissue.[\/vc_column_text][vc_column_text css=”.vc_custom_1490378573383{padding-top: 30px !important;padding-bottom: 30px !important;}”]\"\"[\/vc_column_text][vc_column_text]Fig. 7.<\/strong> Animation showing tensile test using the FEM model of onion epidermis tissue.[\/vc_column_text][vc_column_text]The models showed capabilities of simulating large strains with nonlinear behaviour and produced force-strain curves that closely matched the experimental data. The model revealed a significant influence of tissue structure on micromechanical properties and allowed for interpretation of the tensile test results (force-strain curves) with respect to changes occurring in the structure of the virtual tissue. This proved that qualitative improvement of the results obtained from FEM models of plant tissues is possible due to incorporation of the real microstructure.[\/vc_column_text][vc_column_text css=”.vc_custom_1490378487210{padding-top: 30px !important;padding-bottom: 30px !important;}”]\"\"[\/vc_column_text][vc_column_text]Fig. 8.<\/strong> Experimental stress-strain curve for tensile test of onion epidermis tissue. E1<\/sub> – elasticity modulus in the first part of the curve, E2<\/sub> – modulus of elasticity in second linear part of curve (after transition to elasto-plastic deformation), \u03c3pl<\/sub> and \u03b5pl<\/sub> – stress and strain at the beginning of the transition phase, respectively.[\/vc_column_text][vc_column_text css=”.vc_custom_1490378553353{padding-top: 30px !important;padding-bottom: 30px !important;}”]\"\"[\/vc_column_text][vc_column_text]Fig. 9. <\/strong>FEM models performance for three cell wall parameters: E1exp,<\/sub> E1mod<\/sub> – elasticity modulus in the first part of the curve, E2exp<\/sub>, E2mod<\/sub> – elastic modulus in second linear part of curve, \u03c3pl exp<\/sub>, \u03c3pl mod<\/sub> – stress and at the yielding point.[\/vc_column_text][vc_column_text css=”.vc_custom_1492283284989{padding-top: 30px !important;padding-bottom: 30px !important;}”]Presented model shows innovative approach to modelling of plant micromechanics and the efforts undertaken so far in this field were performed only by a few research institutions worldwide. Although the model has been developed on a relatively simple epidermal tissue, it can be generalized to more complicated tissue with a different shapes of cells and intercellular spaces. Such models are necessary to understand the structure related properties of plants, like fruits and vegetables, and bring, in the next step, possibilities of extension by nano-structural features (like cell wall composition). This knowledge will be useful for engineering and improve fruits and vegetables quality, for instance by growing varieties of fruit, which, due to the morphological characteristics of tissue would be more resistant to postharvest damage or would be developed for specific consumer preferences.[\/vc_column_text][vc_column_text]Future work<\/strong><\/p>\n

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Simulations of fully 3D, automatic generated cellular structures.[\/vc_column_text][vc_column_text css=”.vc_custom_1490378641658{padding-top: 30px !important;}”]\"\"[\/vc_column_text][\/vc_column][\/vc_row]<\/p>\n<\/div>","protected":false},"excerpt":{"rendered":"

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