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FE modelling of Wood Structure Petr Koňas 3.června 2008, Dolní Maxov Ing. Petr Koňas, Ph.D.

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Programy a algoritmy numerické matematiky 14 strana 2/33 Guidelines Why to model wood structure? Problems of geometry –Disadvantages of homogenized models. –Problems in modelling of biol.structures. –Problem of anisotropy. –How deep? Low limits in modelling. –Problem of variability. –Probable structure – probable results. –Accuracy –How far? Up limits in modelling. –What a time? –Advantages and disadvantages of L-systems Coupled physical task Weak solution with mixed multiscaled finite elements

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Programy a algoritmy numerické matematiky 14 strana 3/33 Why model wood structure? - Homogenized models fails - Common models may be too complicated for accurate solution - Although the physical model exists on lower scale, it can be difficult to find appropriate solution on higher scale(s) - Microscopic and lower scaled structures became more important than before mainly for nanotechnology and material science - Evaluation of material model “ab initio” may be more efficient than expensive experiments - Wood and several other parts are structures with remodelling character and its development in time (growing) is closely connected to micro and meso-scales. - Modelling on lower scales can help us to understand more the substance and reasons of final behaviour of wood.

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Programy a algoritmy numerické matematiky 14 strana 4/33 Disadvantages of homogenized models - Reduce information with local character, even though it can influence higher scales - Wood is anisotropic material with high variability. It is difficult to determine appropriated reference region. - Material properties embody variability according to scales. Accuracy of solution on some scale level can be very poor in extension on lower or higher scales. Also accuracy on the same scale can be far off constant character. - When some reference region is chosen, the problem of material properties is still not solved.

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Programy a algoritmy numerické matematiky 14 strana 5/33 Generally insufficient amount of input parameters and verifying experiments. How can we simply describe geometry of biol. structures? How to describe variability of material properties (according to structure, scale, environment)? How to describe relationship of material and geometry properties on scale? How to determine influence of properties from specific scale on behaviour on higher scales? (vegetation × abiotic factors, stand × fluid flow, tree × bush, crown/roots × stem, heartwood × sap, early × late wood, tracheids × parenchyma cells, lumen × cell layers, lignin × cellulose, covalent × ion coupling, quantum effects) How to involve “individuality” and “stereotype”? (selfsimilar/fractal structure × individual occurrence/mutation) How to involve genetic predetermination and phenotypic resignation? Problems in modelling of biol.structures

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Programy a algoritmy numerické matematiky 14 strana 6/33 Problem of anisotropy Anisotropic homogenized models of wood. Simple geometry allows declaring of complex constitutive relationships FE model of anatomy structure of early/late wood. Detail geometry allows to determine influence of small structures (pits) on global behaviour. Complex physical model can be used. Early & late wood FE models include variability on wider range, but complexity of physics has to be lower. FE model of homogenized micro structure (superelements) tends to be sufficient estimation of material on macro scale. (time-consuming)

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Programy a algoritmy numerické matematiky 14 strana 7/33 How deep? Early and late type of tracheid Early and late wood transition Cell wall with layered elements Orientation of fibrils in individual layers

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Programy a algoritmy numerické matematiky 14 strana 8/33 Problem of variability Probabilistic structure How to determine structure of wood for arbitrary position? If statistical behaviour of morphological parameters can be determined, the cross-scale relationships (transition functions) can be evaluated.

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Programy a algoritmy numerické matematiky 14 strana 9/33 Near the stem pith Far off the stem pith Shear RLShear RTShear TL etc. Probable structure – probable results Tension x Compression

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Programy a algoritmy numerické matematiky 14 strana 10/33 Accuracy? Accuracy and large amount of material characteristics provide “accurate” results on statistical sense. “Accuracy” can be also in 30% of variation coefficient. This problem origins from assumption, that geometry has stochastic character (this assumption coincides with the similar statement that material properties has stochastic character), but geometry is consequence of developing stage (growing) and material in every position in stem is not independent, but it is related to position, environment, conditions, etc. Probabilistic FE model are unsuitable for homogenization and solution on higher scales. Solution of such problem can be evolutionary models.

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Programy a algoritmy numerické matematiky 14 strana 11/33 How far? A=aaSaaSaaSaaSaaSaaSaaSaaSa aSaaB S='(.9)!(.9) a=tF[&'(.8)!B|z]>(137)[z&'(.7)!B| z]>(137) B=tF[-'(.8)!(.9)$C|z]'(.9)!(.9)C C=tF[+'(.8)!(.9)$B|z]'(.9)!(.9)B A=F[&'(.7)!B]>(137)[&'(.6)!B] >(137)'(.9)!(.9)A B=F[-'(.7)!(.9)$C]'(.9)!(.9)C C=F[+'(.7)!(.9)$B]'(.9)!(.9)B c(12)FFFFFFFFFF>(1)&(1)A A=!(.75)t(.9)FB>(94)B>(132)B B=[&"t(.9)!(.75)F[|z]$A[|z]] c(12)A A=aaSaaSA S='(.8)!(.9) a=tF[&'(.8)!B]>(137)[z&'(.7)!B]>(137) B=tF[-'(.8)!(.9)$C]'!(.9)C C=tF[+'(.8)!(.9)$B]'!(.9)B c(12)FFFFFA A=!(.8)tFB>(94)C>(132)D B=[&'t(.5)!(.9)F$A|z] C=[&'t(.4)!(.8)F$A|z] D=[&'t(.3)!(.7)F$A|z] Orthotropic material properties, gravitational acceleration, non axis force, base of stem is fixed, 840-1500 geometric components, 100 000-1 200 000 finite elements, 5 mil.DOFs. Lindenmayer grammar allows to create iterated structures with selfsimilar/fractal character as on micro scale as on macro scale. With FE translator very complex geoms can be numerically solved.

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Programy a algoritmy numerické matematiky 14 strana 12/33 What a time? A = !(.8)tFB>(94)C>(132)D B = [&'t(.5)!(.9)F$AL|zL] C = [&'t(.4)!(.8)F$AL|zL] D = [&'t(.3)!(.7)F$AL|zL] L = [~c(8){+(30)f(.5)-(120)f(.5)-(120)f(.5)}] A = aaSaaSaaSaaSaaSaaSaaSaaSaaSaaB S = '(.9)!(.9) a = tF[&'(.8)!LBL|zL]>(137)[z&'(.7)!LBL|zL]>(137) B = tFL[-'(.8)!(.9)$LCL|zL]'(.9)!(.9)C C = tFL[+'(.8)!(.9)$LBL|zL]'(.9)!(.9)B L = [~c(8){+(30)f(.4)-(120)f(.4)-(120)f(.4)}] L-systems Also evolutionary models can be analyzed by Lindenmayer systems

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Programy a algoritmy numerické matematiky 14 strana 13/33 Advantages and disadvantages of L-systems Advantages - L-systems seems to be very realistic (according to geometry and description of material properties distribution) - Models derived by iterated functions include less unknowns and offers tool for evaluation of variables which depends on geometry (permeability, variation of properties) - Model assembling can be less data-intensive - Predictive abilities can be higher in comparison with homogenized models Disadvantages - L-system based models translated finite element mesh are still very huge - Formed models are usually very complicated - Iterated function for geometry of structure has to be revealed. - Solution is very difficult for coupled physical problems - It is multispecialty, money and time consuming problem

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Programy a algoritmy numerické matematiky 14 strana 14/33 Coupled physical task One of the most difficult problem in Wood Science is modelling of microwave wood drying process including micro-effects on small scales. q abs is density of energy, is angular velocity (s -1 ), ‘’ eff is effective relative loss factor, E is electric field (V.m -1 ) B is the magnetic flux density, D is electric flux density, H is magnetic field intensity, J is current density, r e is electric charge density. Due to anisotropy of wood we can itemize these variables to, where e is permittivity, is permeability and is electric conductivity of material.

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Programy a algoritmy numerické matematiky 14 strana 15/33 Coupled physical task When only conduction due to the microwave heating source is considered: When also convective source plays important role: Temperature is not only one of important fields, which is changing during drying process

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Programy a algoritmy numerické matematiky 14 strana 16/33 Coupled physical task Rapid change of moisture, temperature are reasons for large time-depended stress-strain effect. el c is immediate elastic strain (structural), vel c is viscous-elastic part of strain (structural), F() is function of memory effect, D is matrix of elasticity, is relaxation time Both strains are composed from pure mechanical, thermal and moisture components In our case the problem was simplified for visco-elastic strains or for constant memory effect

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Programy a algoritmy numerické matematiky 14 strana 17/33 Coupled physical task Elastic components has usual linear character Also elastic mechanical properties for pure mechanical behaviour should include linear influence of moisture and temperature

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Programy a algoritmy numerické matematiky 14 strana 18/33 Coupled physical task Than modified matrix of elasticity can be formed In this declaration we used k D as constant with the following meaning and vector for simple substitution during separation. Matrix of elasticity can be simply defined: withal Final stress-strain relation on modified matrixes of elasticity

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Programy a algoritmy numerické matematiky 14 strana 19/33 Coupled physical task Previous equation has to be disassembled into unknown displacement by the common relationships. u is vector of displacements, F i are components of volume forces

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Programy a algoritmy numerické matematiky 14 strana 20/33 The final relationship for stress-strain components according to unknown variable displacement u respective u vel can be formed in this grouped form.

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Programy a algoritmy numerické matematiky 14 strana 21/33 Coupled physical task Described model is valid for diffusive transport of moisture and temperature. It is not appropriate (due physical nature of phenomenon) for free water movement. This transport is allocated into intercellular spaces and cell lumen. Description of this process can be done with Navier-Stokes equation. Finally the following set of PDE’s has to be solve:

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Programy a algoritmy numerické matematiky 14 strana 22/33 The weak form of thermal-moisture displacements can be written as follows: for all and meaning of as scalar product on Hilbert space. Let us assume the region is partitioned by linear meshon very fine scalealso we will assume that region is not of small regions are covered by mesh on this scale (subgrids)., where are Raviart-Thomas (RT) spaces. fully partitioned by this fine mesh. Only Functional is than defined on vector subspaces Subspaces may not fill the full space V. It means that Withal we declare mentioned vector subspaces with bases Complete basis on vector space Similarly let us to partition by next linear meshes for different scales where again regions cover some parts of on specific scale. Consequently similar vector subspaces can be distinguished with the same requirements: Weak solution

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Programy a algoritmy numerické matematiky 14 strana 23/33

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Programy a algoritmy numerické matematiky 14 strana 24/33 Weak solution All unknowns can be decomposed to individual scales e.g.: Decomposition of unknowns to individual scales affects solution in sense of finite elements and minimisation of functional ( 1 ) does not provide common appearance of Ritz system Let us consider PDE with differential operator A and follow common steps in solution of this task for multi-scale problem. Functional which will be minimized has standard form: Decomposed unknown will be substituted into first part of functional: It can be expanded due to rules of scalar product in the following manner. As usual the functional is minimized by the function: For first step we will approximate functional in subgrid on scale Finally unknown function can be by this function:

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Programy a algoritmy numerické matematiky 14 strana 25/33 Evaluation L u for minimizing function can be done on these relationships: Requirement on minimisation of quadratic functional F u allows evaluating a minimum of function. Thus partial differentiation according to all coefficients on all scales should be done. This task can be easily achieved by next relations. is modified lower triangular matrix of Ritz system. is modified upper triangular matrix of Ritz system. is well known matrix of Ritz system

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Programy a algoritmy numerické matematiky 14 strana 26/33 Weak solution Applying of mentioned rules on left part of functional leads to: This complex system can be rewritten in more readable form:

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Programy a algoritmy numerické matematiky 14 strana 27/33 When the full functional is minimized by: The solution of the initial problem can be reached by enumeration of By analogy, the solution of coupled problem with applying of S A derivation can be rewritten. for differential operator For differential operators and function

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Programy a algoritmy numerické matematiky 14 strana 28/33 Weak solution Solution is realized in i consequent steps of solution. In first step the previous equation is formed, whereas results of higher scales are unknown (in Ritz or modified Ritz system). Solution on higher scales in individual nodes can be expressed by mapping of From this step we obtain definitions in some nodes on higher scale(s) which bounds region of element on this solved scale. In the following we calculate the same eq., but on the following higher scale withal some nodes on this scale were strictly derived from previous step. This idea is repeated until the highest scale is reached. Advantage of this type of solution is also that you do not need enumerate results on lower scales, but you can enumerate only results on last scale whereas results on this scale is derived from the low and lower scales. or other appropriate lower scales.

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Programy a algoritmy numerické matematiky 14 strana 29/33 Coupled microwave drying of wood non-scaled problem EMAG taskHeating task

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Programy a algoritmy numerické matematiky 14 strana 30/33 Coupled microwave drying of wood non-scaled problem

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Programy a algoritmy numerické matematiky 14 strana 31/33 Computational sources Numerical simulations are very source-consuming processes. Usable and appropriate models consists of more than 3mil. DOF’s. From this reason Dep. of Wood Science on Mendel University built together with Dep. of Theoretical and experimental electrotechnics on Technical University in Brno cluster for high performance computing. We also participate on national grid project (METACENTRUM) for extensive distributed tasks. Finally the EU grid EGEE for scientific computations became big source for our computing. 16 CPU AMD64 (Dp.WS Mendel Univ. & Dp.TEE Technical Univ.) 500 CPU METACENTRUM (FI MU) Dp.WS Mendel Univ. 2500 CPU (EGEE Grid EU)

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Programy a algoritmy numerické matematiky 14 strana 32/33 Acknowledgment The Research project GP106/06/P363 Homogenization of material properties of wood for tasks from mechanics and thermodynamics (Czech Science Foundation) and Institutional research plan MSM6215648902 - Forest and Wood: the support of functionally integrated forest management and use of wood as a renewable raw material (2005-2010, Ministry of Education, Youth and Sport, Czech Republic) supported this work.

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Programy a algoritmy numerické matematiky 14 strana 33/33 Thank you, for your attention...

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Programy a algoritmy numerické matematiky 14 strana 34/33 J. da Cimrman „Teorie externismu“...okolí není tam, kde si myslíme, že je ale je přesně tam, kde si myslíme, že není… K. Gödel’s statement of God’s existence Everybody has its own truth on some „scale“

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