CN112257305A - Method and device for obtaining ion concentration in concrete structure based on finite element method - Google Patents
Method and device for obtaining ion concentration in concrete structure based on finite element method Download PDFInfo
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Abstract
The invention discloses a method and a device for acquiring ion concentration in a concrete structure based on a finite element method, wherein the method comprises the following steps: acquiring a partial differential equation of a diffusion model of any ion in a concrete structure; obtaining a weak form of the partial differential equation based on the partial differential equation; dividing a solving space domain into units, and acquiring a discrete unit finite element control equation based on the weak form of the partial differential equation; dispersing a solving time domain, and acquiring a matrix iteration solving form of the discrete unit finite element control equation based on the discrete unit finite element control equation; and obtaining the concentration of the ions in the concrete structure based on the time step iterative computation and the matrix iterative solution form. The method adopts the quadrilateral eight-node isoparametric unit to divide and solve the spatial domain, can better simulate the structure of the bending boundary, and realizes the beneficial effect of accurately obtaining the ion concentration in the concrete structure containing the bending boundary shape.
Description
Technical Field
The invention relates to the field of concrete materials, in particular to a method and a device for acquiring ion concentration in a concrete structure based on a finite element method.
Background
Sulfate attack is an important factor causing the degradation of the durability of concrete materials and the reduction of the service performance of structures. In recent years, with the rapid development of national economy of China, the construction scale of infrastructure projects is unprecedentedly huge, and the infrastructure projects are closely related to concrete materials and structures. However, the distribution of the service environment of the sulfate type engineering in China is very wide, sulfate ions in the environment diffuse into the concrete structure to corrode the concrete material, so that the mechanical property of the concrete structure is reduced, and the structure fails in advance. Therefore, it is very important to study the diffusion process of sulfate ions in concrete structures to analyze the problem of degradation of concrete durability caused by sulfate attack.
The sulfate ions diffused into the concrete structure can be divided into two parts, one part is freely distributed in the concrete structure in a free state, and the other part can generate chemical reaction with a cement hydration product in the concrete and be consumed. Thus, the diffusion behavior of sulfate ions in concrete is an unsteady state process. According to Fick's second law and the law of conservation of mass, a sulfate ion diffusion model can be established that takes into account the consumption of chemical reactions.
At present, the finite difference method is widely applied to numerical solution of the ion diffusion model. In principle, the finite difference method is arbitrary for grid division of the solution area, however, in practical calculation, a completely regular grid division mode is generally adopted, so that the differential equations at each discrete point in the solution area are in the same form, thereby improving the solution accuracy of the model. The single grid division mode corresponding to the finite difference method has poor adaptability to the complex geometric shape boundary, and is difficult to meet the requirements of the shape of a service concrete structure (such as a cylinder/an elliptic cylinder).
Therefore, the above prior art has at least the following technical problems: in the prior art, the method for acquiring the ion concentration in the concrete structure based on the finite element method adopts a single grid division mode, and has poor adaptability to the boundary with a complex geometric shape, so that the method is not suitable for the concrete structure with the curved boundary shape.
Disclosure of Invention
The embodiment of the application provides a method and a device for acquiring ion concentration in a concrete structure based on a finite element method, and solves the technical problem that the method for acquiring ion concentration in a concrete structure based on a finite element method in the prior art is not suitable for a concrete structure containing a curved boundary shape due to poor adaptability to a complex geometric shape boundary because a single grid division mode is adopted. The embodiment of the application adopts quadrilateral eight-node isoparametric units to divide and solve the spatial domain, can better simulate the structure of the curved boundary, can be used for researching the time-varying law of the sulfate ion concentration in a complex geometric spatial region, and realizes the beneficial effect of accurately obtaining the ion concentration in the concrete structure containing the curved boundary.
In order to solve the above technical problem, in a first aspect, an embodiment of the present application provides a method for obtaining an ion concentration in a concrete structure based on a finite element method, including:
acquiring a partial differential equation of a diffusion model of any ion in a concrete structure;
obtaining a weak form of the partial differential equation based on the partial differential equation;
dividing a solving space domain into units, and acquiring a discrete unit finite element control equation based on the weak form of the partial differential equation;
dispersing a solving time domain, and acquiring a matrix iteration solving form of the discrete unit finite element control equation based on the discrete unit finite element control equation;
and obtaining the concentration of the ions in the concrete structure based on the time step iterative computation and the matrix iterative solution form.
Further, the ion is a sulfate ion.
Further, the obtaining of the partial differential equation of the diffusion model of any one kind of ions in the concrete structure specifically includes:
based on the consumption of the ions by chemical reaction, establishing a diffusion model of the sulfate ions in the concrete structure, wherein the expression of the diffusion model of the sulfate ions in the concrete structure is as follows:
wherein the content of the first and second substances,
kvis the chemical reaction rate in mol/(m)3·s)；cCaIs the concentration of calcium ions formed by the decomposition of cement hydration products in the concrete structure, and the unit is mol/m3Let J be-kvcCa(ii) a u is the concentration of sulfate ions in the concrete structure and the unit is mol/m3(ii) a t is any service time of the concrete structure in a sulfate service environment, and the unit is s (second); x is a horizontal axis coordinate on the cross section of the concrete structure, and the unit is m; y is the coordinate of the longitudinal axis on the cross section of the concrete structure and has the unit of m; (x, y) represents position coordinates within a cross-section of the concrete structure; u. ofdThe concentration of sulfate ions consumed by chemical reaction in the concrete structure is mol/m3(ii) a D is the effective diffusion coefficient of sulfate ions in the concrete structure, and the unit is m2S; gamma is the boundary of the solving space domain of the cross section of the concrete structure, omega is the solving space domain of the cross section of the concrete structure, u0The concentration of sulfate solution in the service environment of the concrete structure is expressed in mol/m3；
The partial differential equation for obtaining the diffusion model of the sulfate ions in the concrete structure is as follows:
further, the weak form of the partial differential equation is as follows:
where v is any function of the coordinates (x, y) in the solution space domain Ω, and v is a function of v,
And
can be integrated, and v equals 0 at the solution space domain boundary Γ, i.e.
Further, the unit division of the solution space domain and the acquisition of the discrete unit finite element control equation based on the weak form of the partial differential equation are specifically as follows:
carrying out grid division on the solved space domain by adopting a quadrilateral eight-node unit so as to carry out unit division on the solved space domain to obtain the discrete units, wherein the obtained finite element control equation of the discrete units is as follows:
wherein e is a discrete unit, WeIs a matrix of cell masses, KeIs a matrix of cell stiffness, ueIs a unit of the ion concentration vector, NeIs a unit shape function vector, BeIs a cell strain matrix.
Further, the matrix iterative solution form of the discrete element finite element control equation is as follows:
further, before obtaining the concentration of the ions in the concrete structure based on the time step iterative computation and the matrix iterative solution form, the method further includes: and correcting the matrix iterative solution form based on the boundary condition, wherein the corrected matrix iterative solution form is as follows:
in the formula (I), the compound is shown in the specification,
for a corrected time point tk+1The concentration vector of sulfate radical ions in the concrete structure,
for a corrected time point tkSulfate ion concentration vector in the concrete;
the corrected integral iteration matrix is obtained; a is a known vector, u, related to the spatially resolved domain boundary ion concentrationΓThe space is solved for sulfate ion concentration vectors at the domain boundaries.
In a second aspect, the present application further provides an apparatus for obtaining an ion concentration in a concrete structure based on a finite element method, the apparatus including:
the first acquisition component is used for acquiring partial differential equations of a diffusion model of any ions in the concrete structure;
second acquiring means for acquiring a weak form of the partial differential equation based on the partial differential equation;
the third acquisition component is used for carrying out unit division on a solved space domain and acquiring a discrete unit finite element control equation based on the weak form of the partial differential equation;
the third acquisition component is used for discretizing a solving time domain and acquiring a matrix iteration solving form of the discrete unit finite element control equation based on the discrete unit finite element control equation;
a fifth obtaining component configured to obtain a concentration of the ions in the concrete structure based on a time-step iterative calculation and the matrix iterative solution form.
In a third aspect, an embodiment of the present application further provides an apparatus for obtaining an ion concentration in a concrete structure based on a finite element method, including a memory, a processor, and a computer program stored in the memory and executable on the processor, where the processor implements the following steps when executing the program:
acquiring a partial differential equation of a diffusion model of any ion in a concrete structure;
obtaining a weak form of the partial differential equation based on the partial differential equation;
dividing a solving space domain into units, and acquiring a discrete unit finite element control equation based on the weak form of the partial differential equation;
dispersing a solving time domain, and acquiring a matrix iteration solving form of the discrete unit finite element control equation based on the discrete unit finite element control equation;
and obtaining the concentration of the ions in the concrete structure based on the time step iterative computation and the matrix iterative solution form.
In a fourth aspect, an embodiment of the present application further provides a computer-readable storage medium, on which a computer program is stored, where the computer program, when executed by a processor, implements the following steps:
acquiring a partial differential equation of a diffusion model of any ion in a concrete structure;
obtaining a weak form of the partial differential equation based on the partial differential equation;
dividing a solving space domain into units, and acquiring a discrete unit finite element control equation based on the weak form of the partial differential equation;
dispersing a solving time domain, and acquiring a matrix iteration solving form of the discrete unit finite element control equation based on the discrete unit finite element control equation;
and obtaining the concentration of the ions in the concrete structure based on the time step iterative computation and the matrix iterative solution form.
One or more technical solutions provided in the embodiments of the present application have at least the following technical effects or advantages:
the embodiment of the application provides a method and a device for acquiring ion concentration in a concrete structure based on a finite element method, wherein the method comprises the following steps: acquiring a partial differential equation of a diffusion model of any ion in a concrete structure; obtaining a weak form of the partial differential equation based on the partial differential equation; dividing a solving space domain into units, and acquiring a discrete unit finite element control equation based on the weak form of the partial differential equation; dispersing a solving time domain, and acquiring a matrix iteration solving form of the discrete unit finite element control equation based on the discrete unit finite element control equation; and obtaining the concentration of the ions in the concrete structure based on the time step iterative computation and the matrix iterative solution form. The method is used for solving the technical problem that the method for acquiring the ion concentration in the concrete structure based on the finite element method in the prior art is not suitable for the concrete structure with the curved boundary shape due to the fact that the method adopts a single grid division mode and has poor adaptability to the boundary with the complex geometric shape. The space domain is divided and solved by adopting quadrilateral eight-node isoparametric units, the structure of the curved boundary can be well simulated, the method can be used for researching the time-varying law of the sulfate ion concentration in the complex geometric space region, and the beneficial effect of accurately obtaining the ion concentration in the concrete structure containing the curved boundary shape is realized.
Drawings
FIG. 1 is a schematic flow chart illustrating a method for obtaining an ion concentration in a concrete structure based on a finite element method according to an embodiment of the present invention;
FIG. 2 is a schematic structural diagram of an overall coordinate system of a method for obtaining ion concentration in a concrete structure based on a finite element method according to an embodiment of the present invention;
FIG. 3 is a schematic diagram of a local coordinate system of the method for obtaining the ion concentration in the concrete structure based on the finite element method in the embodiment of the present invention;
FIG. 4 is a schematic structural diagram of an apparatus for obtaining ion concentration in a concrete structure based on a finite element method according to an embodiment of the present invention;
fig. 5 is a schematic structural diagram of an apparatus for obtaining ion concentration in a concrete structure based on a finite element method according to an embodiment of the present invention.
Description of reference numerals: a first acquisition unit 11, a second acquisition unit 12, a third acquisition unit 13, a fourth acquisition unit 14, a fifth acquisition unit 15, a bus 300, a receiver 301, a processor 302, a transmitter 303, a memory 304, a bus interface 306.
Detailed Description
The embodiment of the application provides a method and a device for acquiring ion concentration in a concrete structure based on a finite element method, and solves the technical problem that the method for acquiring ion concentration in a concrete structure based on a finite element method in the prior art is not suitable for a concrete structure containing a curved boundary shape due to poor adaptability to a complex geometric shape boundary because a single grid division mode is adopted.
In order to solve the technical problem of the crosstalk, the technical scheme in the embodiment of the present application has the following general idea:
the embodiment of the application provides a method and a device for acquiring ion concentration in a concrete structure based on a finite element method, wherein the method comprises the following steps: acquiring a partial differential equation of a diffusion model of any ion in a concrete structure; obtaining a weak form of the partial differential equation based on the partial differential equation; dividing a solving space domain into units, and acquiring a discrete unit finite element control equation based on the weak form of the partial differential equation; dispersing a solving time domain, and acquiring a matrix iteration solving form of the discrete unit finite element control equation based on the discrete unit finite element control equation; and obtaining the concentration of the ions in the concrete structure based on the time step iterative computation and the matrix iterative solution form. The method is used for solving the technical problem that the method for acquiring the ion concentration in the concrete structure based on the finite element method in the prior art is not suitable for the concrete structure with the curved boundary shape due to the fact that the method adopts a single grid division mode and has poor adaptability to the boundary with the complex geometric shape. According to the embodiment of the application, the space domain is divided and solved by adopting the quadrilateral eight-node isoparametric units, the structure of the curved boundary can be well simulated, the method can be used for researching the time-varying law of the concentration of the sulfate ions in the complex geometric space region, and the beneficial effect that the ion concentration in the concrete structure containing the curved boundary shape can be accurately obtained is achieved.
In order to better understand the technical solution, the technical solution will be described in detail with reference to the drawings and the specific embodiments.
Example one
Fig. 1 is a schematic flow chart of a method for obtaining ion concentration in a concrete structure based on a finite element method in an embodiment of the present invention, as shown in fig. 1, the method includes the following steps:
step 110: and acquiring a partial differential equation of a diffusion model of any ions in the concrete structure.
Further, the ions are sulfate ions, chloride ions, calcium ions or the like which diffuse into the concrete structure from a service environment during service of the concrete structure, and in this embodiment, the ions are sulfate ions.
Specifically, the step 110 specifically includes the following steps:
step 111: based on Fick's second law and mass conservation law, and considering the consumption of chemical reaction to sulfate ion, establishing a diffusion model of sulfate ion in concrete structure, and the expression of the diffusion model of sulfate ion in concrete structure is:
in the formula (1), the reaction mixture is,
kvis the chemical reaction rate in mol/(m)3·s)；cCaIs the concentration of calcium ions formed by the decomposition of cement hydration products in the concrete structure, and the unit is mol/m3Let J be-kvcCa(ii) a u is the concentration of sulfate ions in the concrete structure and the unit is mol/m3(ii) a t is any service time of the concrete structure in a sulfate service environment, and the unit is s (second); x is a horizontal axis coordinate on the cross section of the concrete structure, and the unit is m; y is the coordinate of the longitudinal axis on the cross section of the concrete structure and has the unit of m; (x, y) represents position coordinates within a cross-section of the concrete structure; u. ofdThe concentration of sulfate ions consumed by chemical reaction in the concrete structure is mol/m3(ii) a D is the effective diffusion coefficient of sulfate ions in the concrete structure, and the unit is m2S; gamma is the boundary of the solving space domain of the cross section of the concrete structure, omega is the solving space domain of the cross section of the concrete structure, u0The concentration of sulfate solution in the service environment of the concrete structure is expressed in mol/m3。
Specifically, the solved spatial domain Ω refers to the cross section of the concrete structure selected for sulfate ion concentration measurement; the solved space domain boundary Γ refers to the boundary of the cross section of the concrete structure.
Step 112: obtaining a partial differential equation of the diffusion model of sulfate ions in the concrete structure, which comprises the following specific steps:
step 120: a weak form of the partial differential equation is obtained based on the partial differential equation.
Specifically, the partial differential equation (2) of the sulfate ion diffusion model in the concrete structure is converted into a weak form by using a Galerkin (Galerkin) method, and the weak form is as follows:
in equation (3), v is an arbitrary function with respect to the coordinates (x, y) in the solution space domain Ω, and v is required to satisfy v, and v in the solution space domain Ω,
And
can be integrated, and v equals 0 at the solution space domain boundary Γ, i.e.
Step 130: and carrying out unit division on the solution space domain, and acquiring a discrete unit finite element control equation based on the weak form of the partial differential equation.
The step 130 specifically includes:
adopting a quadrilateral eight-node parameter unit to perform unit division on the solved spatial domain omega to obtain a discrete unit e, and dividing parameter units such as a quadrilateral eight-node of the solved spatial domain omega, wherein the quadrilateral eight-node parameter unit can construct a structure with a complex geometric boundary as shown in FIGS. 2 and 3; specifically, the triangle units generally adopted in the finite element method have good adaptability and high convergence of calculation results, but the precision is not ideal enough and cannot approach the shape of the boundary of a curve well; therefore, the quadrilateral eight-node parameter unit is adopted in the embodiment of the application, and the shape of the boundary of the curve can be better approximated. And (3) carrying out coordinate system conversion, transferring the unit e in the global coordinate system (x-y) to a local coordinate system (xi-eta) (wherein xi is the horizontal axis coordinate of the local coordinate system, and eta is the vertical axis coordinate of the local coordinate system), and the corresponding shape function vector in the local coordinate system is as follows:
in the formulae (4) and (5), NeAs a vector of the shape function of said discrete unit e, Ni(xi, η) is a shape function at the discrete unit node i under the local coordinate system; (xi)i,ηi) Is the coordinate of the discrete unit node i under the local coordinate system, and xiiIs the abscissa, eta, of the discrete unit node i in a local coordinate systemiThe vertical coordinate of the discrete unit node i under the local coordinate system; and the coordinates of the eight nodes are (-1, -1), (1,1), (-1,1), (0, -1), (1,0), (0,1) and (-1,0) in sequence.
Therefore, the coordinate x at any position in the discrete unit e in the global coordinate system (x-y) can be expressed as:
in the formula (6), xi＝{xi yiAnd the coordinates of the discrete unit node i under the integral coordinate system are obtained.
Further, it is possible to obtain:
note the book
Is the strain matrix of the discrete element e, then equation (7) can be expressed as
In the formula (8), J is a jacobian matrix, and T represents a transpose of the matrix.
Therefore, the first term on the left of the equation for the diffusion model weak form (3) can be converted to discrete unit form as follows:
in the formula (9), veElement vector, u, being an arbitrary function veIs the unit vector of the sulfate ion concentration u; note We＝∫∫ΩNe(Ne)Tdxdy is the cell quality matrix, then equation (9) can be simplified as:
the second term on the left of the equation in the form of the diffusion model weak equation can be converted to the discrete unit form:
note Ke＝∫∫ΩBe(Be)Tdxdy is the cell stiffness matrix, then equation (11) can be simplified to
Is obtained from
Since for any veIf all the equations (13) are satisfied, the following equation is satisfied:
in the formula, WeIs a matrix of cell masses, KeIs a matrix of cell stiffness, ueIs a unit of the ion concentration vector, NeIs a unit shape function directionAmount, BeIs a cell strain matrix.
The above formula (14) is a discrete unit finite element control equation of the sulfate ion diffusion model in the concrete structure.
Step 140: and dispersing the solution time domain, and acquiring a matrix iteration solution form of the finite element control equation of the discrete unit.
Specifically, the solution time domain T is a service life of the concrete structure in a sulfate service environment, for example, the service life of the concrete structure is 100000 seconds, and the solution time domain T is [0,100000 ].
Further, the discretizing of the solving time domain T specifically includes:
performing discrete processing on the solving time domain T, and when the time step is delta T, the time step number is K which is T/delta T; therefore, the time domain T is discrete to a time point (0 ═ T)0＜t1＜...＜tk＜...＜tK＝T)。
For said time of service t ∈ (t)k,tk+1) Converting t into a variable
Then time step (t)k,tk+1) Concentration vector u of sulfate ion in inner unite(t) can be expressed as:
in the formula (15), the reaction mixture is,
and
respectively at a time point tkAnd tk+1And (3) obtaining a sulfate ion concentration vector of the time unit:
thus, the discrete element finite element control equations may be converted to
We(ue,k+1-ue,k)＝(JΔtWe-ΔtKe)[(1-τ)ue,k+τue,k+1] (17)
Multiplying both the left and right sides of the equation of the above formula (17)
Order to
Can be simplified to obtain:
(We-JΔtΘWe+ΔtΘDKe)ue,k+1＝[We+(1-Θ)JΔtWe-(1-Θ)ΔtDKe]ue,k (18)
in the formula (18), ω (τ) is an arbitrary function with respect to the variable τ, and satisfies
And Θ, taken as ω (τ) ═ 1, then equation (18) above can be expressed as:
in the formula (19), HeFor the element iteration matrix, equation (19) is the matrix iteration solution form of the discrete element finite element control equation.
Step 150: and obtaining the concentration of the ions in the concrete structure based on the time step iterative computation and the matrix iterative solution form.
Further, before the step 150, the method further includes: correcting the matrix iterative solution form based on the boundary condition, wherein the correction of the matrix iterative solution form based on the boundary condition specifically comprises the following steps:
according to the corresponding relation of the unit node numbers under the local coordinate system and the global coordinate system, the unit iteration matrix HeAssembling the iteration matrix to an integral iteration matrix H; likewise, the unit sulfate ion concentration vector ue,k+1And ue,kExpansion into a global concentration vector uk+1And uk(ii) a According to the boundary conditions of the diffusion model, namely the boundary conditions: u < u > C(x,y)∈Γ＝u0And correcting the whole matrix H and the whole concentration vector u, so as to obtain a modified matrix iterative solution form:
in the formula (20), the reaction mixture is,
for a corrected time point tk+1The concentration vector of sulfate radical ions in the concrete structure,
for a corrected time point tkSulfate ion concentration vector in the concrete;
the corrected integral iteration matrix is obtained; a is a known vector, u, related to the spatially resolved domain boundary ion concentrationΓThe space is solved for sulfate ion concentration vectors at the domain boundaries.
Then, according to the known concentration vector u of the initial time point1(comprises
And uΓOf) and the modified matrix iterative solution form (20), i.e.
A point of time t can be obtained2Concentration vector u of sulfate ion in concrete2(ii) a By means of the iterative calculation, the sulfate ion concentration of any position in the concrete structure at different time points can be obtained.
Further, computer analysis programs for realizing the steps 110-150 are compiled through MATLAB software, so that the concentration of sulfate ions in the concrete structure at different service moments can be automatically obtained.
Specifically, the MATLAB software can perform matrix operations, draw functions and data, implement algorithms, create user interfaces, interface programs in other programming languages, and the like. The basic data unit of MATLAB is matrix, its instruction expression is very similar to the form commonly used in mathematics and engineering, so that it uses MATLAB to implement step 110-step 150 to program computer analysis program to implement automatic batch solution of sulfate radical ion concentration in different positions in concrete structure at different service time.
The steps 110-150 are realized by compiling a computer analysis program through MATLAB software, and for different conditions of concrete corrosion by sulfate, such as ambient temperature, concentration of sulfate solution in the environment, performance of concrete material, shape and size of a concrete structure, the computer analysis program can be realized by utilizing the program for calculation only by modifying corresponding parameter parts in the subprogram.
In summary, embodiments of the present application provide a method for obtaining an ion concentration in a concrete structure based on a finite element method, the method including: acquiring a partial differential equation of a diffusion model of any ion in a concrete structure; obtaining a weak form of the partial differential equation based on the partial differential equation; dividing a solving space domain into units, and acquiring a discrete unit finite element control equation based on the weak form of the partial differential equation; dispersing a solving time domain, and acquiring a matrix iteration solving form of the discrete unit finite element control equation based on the discrete unit finite element control equation; and obtaining the concentration of the ions in the concrete structure based on the time step iterative computation and the matrix iterative solution form. The method is used for solving the technical problem that the method for acquiring the ion concentration in the concrete structure based on the finite element method in the prior art is not suitable for the concrete structure with the curved boundary shape due to the fact that the method adopts a single grid division mode and has poor adaptability to the boundary with the complex geometric shape. The space domain is divided and solved by adopting quadrilateral eight-node isoparametric units, the structure of the curved boundary can be well simulated, the method can be used for researching the time-varying law of the sulfate ion concentration in the complex geometric space region, and the beneficial effect of accurately obtaining the ion concentration in the concrete structure containing the curved boundary shape is realized.
Example two
Based on the same inventive concept as the method for obtaining ion concentration in a concrete structure based on the finite element method in the previous embodiment, the present invention also provides a device for obtaining ion concentration in a concrete structure based on the finite element method, as shown in fig. 4, the device comprises:
a first obtaining part 11, wherein the first obtaining part 11 is used for obtaining partial differential equations of a diffusion model of any kind of ions in the concrete structure;
second acquiring means 12 for acquiring a weak form of the partial differential equation based on the partial differential equation;
the third acquisition component 13 is used for carrying out unit division on a solved space domain and acquiring a discrete unit finite element control equation based on a weak form of the partial differential equation;
a fourth obtaining component 14, where the third obtaining component 14 is configured to discretize a solution time domain, and obtain a matrix iteration solution form of the discrete element finite element control equation based on the discrete element finite element control equation;
a fifth obtaining part 15, wherein the third obtaining part 15 is configured to obtain the concentration of the ions in the concrete structure based on the time step iterative calculation and the matrix iterative solution form.
Further, the apparatus further comprises a first correction component for correcting the matrix iterative solution form based on a boundary condition.
Various modifications and specific examples of the method for obtaining the ion concentration in the concrete structure based on the finite element method in the first embodiment of fig. 1 are also applicable to the apparatus for obtaining the ion concentration in the concrete structure based on the finite element method in the present embodiment, and the method for obtaining the ion concentration in the concrete structure based on the finite element method in the present embodiment is clear to those skilled in the art from the foregoing detailed description of the method for obtaining the ion concentration in the concrete structure based on the finite element method, so for the brevity of the description, the detailed description is omitted here.
EXAMPLE III
Based on the same inventive concept as the method for obtaining the ion concentration in the concrete structure based on the finite element method in the foregoing embodiment, the present invention also provides a device for obtaining the ion concentration in the concrete structure based on the finite element method, on which a computer program is stored, which when executed by a processor implements the steps of any one of the methods for obtaining the ion concentration in the concrete structure based on the finite element method described above.
Where in fig. 5 a bus architecture (represented by bus 300), bus 300 may include any number of interconnected buses and bridges, bus 300 linking together various circuits including one or more processors, represented by processor 302, and memory, represented by memory 304. The bus 300 may also link together various other circuits such as peripherals, voltage regulators, power management circuits, and the like, which are well known in the art, and therefore, will not be described any further herein. A bus interface 306 provides an interface between the bus 300 and the receiver 301 and transmitter 303. The receiver 301 and the transmitter 303 may be the same element, i.e., a transceiver, providing a means for communicating with various other apparatus over a transmission medium.
The processor 302 is responsible for managing the bus 300 and general processing, and the memory 304 may be used for storing data used by the processor 302 in performing operations.
Example four
Based on the same inventive concept as the method of obtaining the ion concentration in the concrete structure based on the finite element method in the foregoing embodiment, the present invention also provides a computer-readable storage medium having stored thereon a computer program which, when executed by a processor, realizes the steps of:
acquiring a partial differential equation of a diffusion model of any ion in a concrete structure; obtaining a weak form of the partial differential equation based on the partial differential equation; carrying out unit division on a solved space domain, and obtaining a discrete unit finite element control equation based on the weak form of the partial differential equation; dispersing a solving time domain, and obtaining a matrix iteration solving form of the discrete unit finite element control equation based on the discrete unit finite element control equation; and obtaining the concentration of the ions in the concrete structure based on the time step iterative computation and the matrix iterative solution form.
In a specific implementation, when the program is executed by a processor, any method step in the first embodiment may be further implemented.
One or more technical solutions in the embodiments of the present application have at least one or more of the following technical effects:
the embodiment of the application provides a method and a device for acquiring ion concentration in a concrete structure based on a finite element method, wherein the method comprises the following steps: acquiring a partial differential equation of a diffusion model of any ion in a concrete structure; obtaining a weak form of the partial differential equation based on the partial differential equation; dividing a solving space domain into units, and acquiring a discrete unit finite element control equation based on the weak form of the partial differential equation; dispersing a solving time domain, and acquiring a matrix iteration solving form of the finite element control equation with the discrete units based on the finite element control equation with the discrete units; and obtaining the concentration of the ions in the concrete structure based on the time step iterative computation and the matrix iterative solution form. The method is used for solving the technical problem that the method for acquiring the ion concentration in the concrete structure based on the finite element method in the prior art is not suitable for the concrete structure with the curved boundary shape due to the fact that the method adopts a single grid division mode and has poor adaptability to the boundary with the complex geometric shape. The space domain is divided and solved by adopting quadrilateral eight-node isoparametric units, the structure of the curved boundary can be well simulated, the method can be used for researching the time-varying law of the sulfate ion concentration in the complex geometric space region, and the beneficial effect of accurately obtaining the ion concentration in the concrete structure containing the curved boundary shape is realized.
As will be appreciated by one skilled in the art, embodiments of the present invention may be provided as a method, system, or computer program product. Accordingly, the present invention may take the form of an entirely hardware embodiment, an entirely software embodiment or an embodiment combining software and hardware aspects. Furthermore, the present invention may take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, and the like) having computer-usable program code embodied therein.
The present invention is described with reference to flowchart illustrations and/or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the invention. It will be understood that each flow and/or block of the flow diagrams and/or block diagrams, and combinations of flows and/or blocks in the flow diagrams and/or block diagrams, can be implemented by computer program instructions. These computer program instructions may be provided to a processor of a general purpose computer, special purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, create means for implementing the functions specified in the flowchart flow or flows and/or block diagram block or blocks.
These computer program instructions may also be stored in a computer-readable memory that can direct a computer or other programmable data processing apparatus to function in a particular manner, such that the instructions stored in the computer-readable memory produce an article of manufacture including instruction means which implement the function specified in the flowchart flow or flows and/or block diagram block or blocks.
These computer program instructions may also be loaded onto a computer or other programmable data processing apparatus to cause a series of operational steps to be performed on the computer or other programmable apparatus to produce a computer implemented process such that the instructions which execute on the computer or other programmable apparatus provide steps for implementing the functions specified in the flowchart flow or flows and/or block diagram block or blocks.
It will be apparent to those skilled in the art that various changes and modifications may be made in the present invention without departing from the spirit and scope of the invention. Thus, if such modifications and variations of the present invention fall within the scope of the claims of the present invention and their equivalents, the present invention is also intended to include such modifications and variations.
Claims (10)
1. The method for acquiring the ion concentration in the concrete structure based on the finite element method is characterized by comprising the following steps:
acquiring a partial differential equation of a diffusion model of any ion in a concrete structure;
obtaining a weak form of the partial differential equation based on the partial differential equation;
dividing a solving space domain into units, and acquiring a discrete unit finite element control equation based on the weak form of the partial differential equation;
dispersing a solving time domain, and acquiring a matrix iteration solving form of the discrete unit finite element control equation based on the discrete unit finite element control equation;
and obtaining the concentration of the ions in the concrete structure based on the time step iterative computation and the matrix iterative solution form.
2. The method for obtaining an ion concentration in a concrete structure based on the finite element method according to claim 1, wherein the ions are sulfate ions.
3. The method for obtaining the ion concentration in the concrete structure based on the finite element method according to claim 2, wherein the partial differential equation for obtaining the diffusion model of any one kind of ions in the concrete structure specifically comprises:
based on the consumption of the ions by chemical reaction, establishing a diffusion model of the sulfate ions in the concrete structure, wherein the expression of the diffusion model of the sulfate ions in the concrete structure is as follows:
wherein the content of the first and second substances,
kvis the chemical reaction rate in mol/(m)3·s)；cCaIs the concentration of calcium ions formed by the decomposition of cement hydration products in the concrete structure, and the unit is mol/m3Let J be-kvcCa(ii) a u is the concentration of sulfate ions in the concrete structure and the unit is mol/m3(ii) a t is any service time of the concrete structure in a sulfate service environment, and the unit is s (second); x is a horizontal axis coordinate on the cross section of the concrete structure, and the unit is m; y is the coordinate of the longitudinal axis on the cross section of the concrete structure and has the unit of m; (x, y) represents position coordinates within a cross-section of the concrete structure; u. ofdThe concentration of sulfate ions consumed by chemical reaction in the concrete structure is mol/m3(ii) a D is the effective diffusion coefficient of sulfate ions in the concrete structure, and the unit is m2S; gamma is the boundary of the solving space domain of the cross section of the concrete structure, omega is the solving space domain of the cross section of the concrete structure, u0The concentration of sulfate solution in the service environment of the concrete structure is expressed in mol/m3；
The partial differential equation for obtaining the diffusion model of the sulfate ions in the concrete structure is as follows:
4. the method of obtaining an ion concentration in a concrete structure based on the finite element method as set forth in claim 2, wherein the weak form of the partial differential equation is as follows:
where v is any function of the coordinates (x, y) in the solution space domain Ω, and v is a function of v,
And
can be integrated, and v equals 0 at the solution space domain boundary Γ, i.e.
5. The method for obtaining the ion concentration in the concrete structure based on the finite element method according to claim 2, wherein the step of dividing the solution space domain into units and obtaining the discrete unit finite element control equation based on the weak form of the partial differential equation comprises:
carrying out grid division on the solved space domain by adopting a quadrilateral eight-node unit, thereby carrying out unit division on the solved space domain to obtain discrete units, wherein the obtained finite element control equation of the discrete units is as follows:
wherein e is a discrete unit, WeIs a matrix of cell masses, KeIs a matrix of cell stiffness, ueIs a unit of the ion concentration vector, NeIs a unit shape function vector, BeIs a cell strain matrix.
6. The finite element method-based method for obtaining the ion concentration in the concrete structure according to claim 2, wherein the matrix iterative solution form of the discrete element finite element control equation is as follows:
7. the method of claim 2, wherein said time step-based iterative computation and said matrix iterative solution form further comprise, prior to obtaining said ion concentration in the concrete structure: and correcting the matrix iterative solution form based on the boundary condition, wherein the corrected matrix iterative solution form is as follows:
in the formula (I), the compound is shown in the specification,
for a corrected time point tk+1The concentration vector of sulfate radical ions in the concrete structure,
for a corrected time point tkSulfate ion concentration vector in the concrete;
the corrected integral iteration matrix is obtained; a is a known vector, u, related to the spatially resolved domain boundary ion concentrationΓThe space is solved for sulfate ion concentration vectors at the domain boundaries.
8. An apparatus for obtaining ion concentration in a concrete structure based on a finite element method, the apparatus comprising:
the first acquisition component is used for acquiring partial differential equations of a diffusion model of any ions in the concrete structure;
second acquiring means for acquiring a weak form of the partial differential equation based on the partial differential equation;
the third acquisition component is used for carrying out unit division on a solved space domain and acquiring a discrete unit finite element control equation based on the weak form of the partial differential equation;
the third acquisition component is used for discretizing a solving time domain and acquiring a matrix iteration solving form of the discrete unit finite element control equation based on the discrete unit finite element control equation;
a fifth obtaining component configured to obtain a concentration of the ions in the concrete structure based on a time-step iterative calculation and the matrix iterative solution form.
9. The device for acquiring the ion concentration in the concrete structure based on the finite element method comprises a memory, a processor and a computer program which is stored on the memory and can run on the processor, and is characterized in that the processor executes the program to realize the following steps:
acquiring a partial differential equation of a diffusion model of any ion in a concrete structure;
obtaining a weak form of the partial differential equation based on the partial differential equation;
dividing a solving space domain into units, and acquiring a discrete unit finite element control equation based on the weak form of the partial differential equation;
dispersing a solving time domain, and acquiring a matrix iteration solving form of the discrete unit finite element control equation based on the discrete unit finite element control equation;
and obtaining the concentration of the ions in the concrete structure based on the time step iterative computation and the matrix iterative solution form.
10. A computer-readable storage medium, on which a computer program is stored, which program, when executed by a processor, carries out the steps of:
acquiring a partial differential equation of a diffusion model of any ion in a concrete structure;
obtaining a weak form of the partial differential equation based on the partial differential equation;
dividing a solving space domain into units, and acquiring a discrete unit finite element control equation based on the weak form of the partial differential equation;
dispersing a solving time domain, and acquiring a matrix iteration solving form of the discrete unit finite element control equation based on the discrete unit finite element control equation;
and obtaining the concentration of the ions in the concrete structure based on the time step iterative computation and the matrix iterative solution form.
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| CN113051794A (en) * | 2021-03-12 | 2021-06-29 | 武汉理工大学 | Method and device for calculating concentration diffusion of impurity elements of quartz and storage medium |
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