CN109871612B - Heterogeneous catalysis surface coverage obtaining method combining ODE integration and Newton method iteration - Google Patents
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- 238000000034 method Methods 0.000 title claims abstract description 52
- 230000010354 integration Effects 0.000 title claims abstract description 43
- 238000007210 heterogeneous catalysis Methods 0.000 title claims abstract description 12
- 238000006243 chemical reaction Methods 0.000 claims abstract description 25
- 238000004422 calculation algorithm Methods 0.000 claims abstract description 22
- 230000003197 catalytic effect Effects 0.000 claims abstract description 19
- 238000004088 simulation Methods 0.000 claims abstract description 11
- 230000008859 change Effects 0.000 claims abstract description 9
- 239000002156 adsorbate Substances 0.000 claims description 5
- 239000000126 substance Substances 0.000 claims description 3
- 238000009396 hybridization Methods 0.000 claims description 2
- 230000008569 process Effects 0.000 description 7
- 230000014509 gene expression Effects 0.000 description 3
- 238000004364 calculation method Methods 0.000 description 2
- 238000009903 catalytic hydrogenation reaction Methods 0.000 description 2
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Abstract
The invention relates to a heterogeneous catalysis surface coverage obtaining method combining ODE integration and Newton method iteration, which comprises the following steps: 1) starting a catalytic simulation; 2) randomly initializing a surface coverage and constructing a micro-dynamics steady-state equation; 3) taking the initialized coverage as a starting point, and integrating the steady-state equation by adopting an ODE numerical integration algorithm; 4) when the integral reaches the designated time, obtaining the current solution as the initial value of Newton iteration and sending the initial value into a Newton iterator for Newton iteration; 5) if the change rate of the coverage along with the time is smaller than the convergence precision, obtaining a steady state solution and performing the step 6), if the change rate of the coverage along with the time is not smaller than the convergence precision, returning to the step 3), taking the current solution as the starting point of an ODE numerical integration algorithm, continuing ODE integration, and repeating the steps 3) -5); 6) and obtaining the reaction rate of the catalytic system according to the steady state solution. Compared with the prior art, the method has the advantages of no excessive dependence on an initial value, memory saving, efficiency improvement, high flexibility and the like.
Description
Technical Field
The invention relates to the field of heterogeneous catalysis micro-dynamics simulation, in particular to a heterogeneous catalysis surface coverage obtaining method combining ODE integration and Newton method iteration.
Background
Micro-kinetic simulation (micro-kinetic simulation) plays a crucial role in connecting micro-scale first-principle calculation with macro-kinetic phenomena, and the method is widely and successfully applied to various researches on complex surface heterogeneous catalysis. At present, researchers can calculate thermodynamic and kinetic energy data of each possible elementary reaction in a catalytic system through commercial quantum chemistry software, and a micro-kinetic method based on the data and on average field approximation can use a computer to quickly and efficiently predict the reaction rate of the catalytic system and other macroscopic properties such as surface composition and elementary reaction reversibility, a speed-decision step and the like.
The micro-kinetic method is a numerical simulation method of the kinetic properties of a chemical reaction system based on mean field approximation and steady state approximation in physical chemistry. A heterogeneous catalytic reaction system is a catalytic cycle consisting of a plurality of elementary reactions. Wherein each elementary reaction can be represented by a general reaction equation:
wherein A isiRepresents an adsorbate, a gas phase molecule or a surface vacancy,
is represented by AiThe coefficients in the elementary reaction e are,
the reaction rate of the elementary reaction can be written by the law of mass action:
wherein the content of the first and second substances,
indicates adsorptionThing AiIn some elementary reactions,
partial pressure of gas phase molecules may also be used
Instead, for a certain adsorbed species, an expression for the rate of change of the surface coverage of the species with time can be derived from the above formula:
for multiple species, the entire catalytic system yields a set of ordinary differential equations:
according to the steady state approximation, when the whole catalytic system reaches the steady state, the species coverage of the surface should be kept in the steady state, i.e. the rate of change of the coverage with time is 0, so that the above ordinary differential equation system can be converted into a set of nonlinear equations:
the coverage of the surface species of the whole catalytic system in a steady state can be obtained by solving the nonlinear equation system or the ordinary differential equation system, and then the reaction rate and other macroscopic properties of the whole system are predicted.
At present, a common method for solving a microscopic kinetic equation is a newton method, which needs to give an initial value (initial coverage) in advance, and then iterates through a newton iterative algorithm, and when the time-dependent change rate of the coverage of all species is close to 0 and is smaller than a certain precision (for example, 10E-50), the whole system can be considered to reach a stable state, and then the iteration can be terminated. However, the iterative solution effect (whether the solution can be converged to sufficient precision and whether the solution after reaching the precision has reasonable physical significance) of the method depends heavily on the value of the initial value of the iteration, and if the value of the initial value is 'unreasonable', the newton method iteration is difficult to obtain the final correct solution. One is that it is difficult for the iterative algorithm to converge the error to a specified accuracy, and the coverage resulting from convergence of low accuracy has a large error that cannot be used as a result; the second case is that newton's method can converge to a specified accuracy, but the resulting solution is not the final reasonable solution of physical significance, e.g., solving for coverage of negative numbers, or coverage that does not satisfy the mass conservation relationship, etc. Especially when the catalytic system required to perform micro-kinetic simulation is complex (including many elementary reactions), the stability and accuracy of solving the steady-state equation are more difficult to guarantee.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provide a heterogeneous catalysis surface coverage acquisition method combining ODE integration and Newton method iteration.
The purpose of the invention can be realized by the following technical scheme:
a heterogeneous catalysis surface coverage obtaining method combining ODE integration and Newton method iteration comprises the following steps:
1) starting a micro-kinetic simulation of the heterogeneous catalysis;
2) randomly initializing a surface coverage and constructing a micro-dynamics steady-state equation;
3) taking the initialized coverage as a starting point, and integrating the steady-state equation by adopting an ODE numerical integration algorithm;
4) when the integral reaches the designated time, terminating ODE numerical integration, and obtaining a current solution as an initial value of Newton iteration to be sent into a Newton iterator for Newton iteration;
5) if the change rate of the coverage along with the time is smaller than the convergence precision, terminating the iteration, obtaining a steady state solution, performing the step 6), if the coverage is not converged, returning to the step 3), taking the current solution as a starting point of an ODE numerical integration algorithm, continuing the ODE numerical integration, and repeating the steps 3) -5);
6) and obtaining the reaction rate of the catalytic system according to the steady state solution.
In the step 2), the micro-dynamics steady-state equation is as follows:
wherein the content of the first and second substances,
is adsorbate AnThe degree of coverage on the surface is,
is adsorbate AnCoefficient in elementary reaction e, reIs the reaction rate of the elementary reaction.
The convergence precision in the step 5) is 10E-50.
In the step 4), the time span and the time step of the ODE numerical integration are automatically adjusted along with the hybrid iteration, and the following steps are included:
T＝α×10N
I＝10-(N+2)
wherein T is the time span of integration, I is the time step length, alpha is an adjustable value with the size of 0-1, and N is the number of times of current hybridization iteration.
In the step 3), an ODE numerical integration algorithm adopts a real-valued variable coefficient ODE integral solver.
The step 4) specifically comprises the following steps:
41) the solution of ODE numerical integration is taken as the initial coverage theta0；
42) Obtaining an iteration direction d:
43) the iteration rate constant lambda is equal to 0.5i-1As an iteration step length, wherein i is the step number of the current Newton iteration;
44) after obtaining the iteration direction and step length, pass Θk+1＝Θk+ λ d update coverage;
45) terminating the iteration when the iteration converges, otherwise returning to step 42) until convergence.
Compared with the prior art, the invention has the following advantages:
the invention adopts the ODE integral value as the input of the initial value of the Newton method, and weakens the excessive dependence of the whole algorithm on the value of the initial value.
And secondly, the ODE integration adopts low-precision numerical representation and has the advantages of saving memory, improving efficiency and the like compared with high-precision floating point numerical representation.
And thirdly, the ODE integration can be used for inputting the integral value as a Newton method and recording the process of surface evolution, and has more information with physical significance than pure Newton method iteration.
And fourthly, the improved hybrid iterative algorithm adopts a strategy of multiple hybrid iterations, even if one iteration cannot obtain a convergent steady-state solution, the algorithm can continuously repeat the ODE/Newton integral switching process, and integral parameters can be dynamically adjusted according to the switching times, so that higher flexibility is improved.
Drawings
FIG. 1 is a flow chart of the method of the present invention.
Fig. 2 is an iterative error evolution curve.
Detailed Description
The invention is described in detail below with reference to the figures and specific embodiments.
As shown in FIG. 1, the invention adopts a hybrid iterative algorithm combining low-precision ODE integration and high-precision Newton method iteration to enable the micro-dynamics simulation of heterogeneous catalysis to overcome the difficulty that a steady-state solution cannot be obtained due to excessive dependence on an initial value. The ODE integration with low precision can be integrated from any initial value, so that the integration is not dependent on the value of the initial coverage, when the integration reaches a certain degree, the coverage of the surface species can evolve gradually along with the time, the ODE integration is stopped at a set time point, the obtained current species coverage value is taken as a rough initial point to be brought into a high-precision Newton iteration solver, and a high-precision steady-state solution is obtained by using a Newton method. If the Newton method still can not be converged, switching to the ODE integrator to continue to use the ODE algorithm to solve the integral, and repeatedly switching between the two algorithms until the change rate of the coverage degree converges to the convergence standard to obtain a steady-state solution.
The brief flow of the algorithm is as follows:
1) starting simulation;
2) randomizing an initial surface coverage to construct a micro-kinetic steady state equation;
3) taking the initialized coverage as a starting point, integrating a steady-state equation by using an ODE numerical integration algorithm, wherein a real-value variable coefficient ODE integration solver is usually used in the integration, and the solver is realized by using fixed-leading-coefficient, so that the implicit Adam method and the method based on a Backward Difference Formula (BDF) can be automatically switched;
4) integrating to a specified time, terminating ODE integration, taking the obtained current solution as an initial value (initial coverage) and carrying out high-precision Newton iteration in a Newton iterator, specifically;
here, the vector function F and the coverage vector Θ are used to represent the steady state equation set, where Θ is the final steady state solution, and the iteration process is as follows:
a) giving an initial coverage Θ 0 (i.e., the previous ODE integration result);
b) obtaining an iteration direction d by solving the following formula;
c) calculating an iteration rate constant lambda being 0.5i-1 as an iteration step, wherein i is the step number of the current Newton iteration;
d) after obtaining the iteration direction and the step length, the step length is determined by thetai+1＝Θi+ λ d to update the coverage vector while making k + 1;
e) if F (theta 0) <, the iteration is converged, and the iteration is terminated; otherwise, returning to b) and repeating the process;
5) if the change rate of the coverage along with the time is smaller than the convergence standard, terminating the iteration; if not, returning to the step 3), taking the current solution as a starting point, entering ODE integration, and repeating the steps;
6) and (4) carrying out data post-processing, and predicting other macroscopic properties such as the reaction rate of the catalytic system based on steady state solution.
Example (b):
the invention simulates the micro-dynamics of the CO2 catalytic hydrogenation process by water assistance on the surface of Cu (211)
Table 1 shows elementary reaction expressions and energy data of a certain path of water-assisted CO2 catalytic hydrogenation process on Cu (211), which are obtained by quantum chemical software VASP calculation. Therefore, the events of the whole process can be substituted into different micro dynamics solving methods for dynamics simulation and comparison.
TABLE 1 all possible elementary reaction expressions and energy data
For comparison, a standard random initial value Newton method and an improved ODE/Newton method hybrid iterative algorithm are respectively used for solving a micro dynamic model of the same catalytic system, and an obtained Newton method iterative error evolution curve is shown in FIG. 2.
As can be seen from FIG. 2, the hybrid iterative algorithm converges to the power of 10E-100 after one ODE/Newton iteration, and the final result is verified to be a final solution with reasonable physical significance. However, in the standard newton iterative algorithm initialized randomly, it is still difficult to converge to a correct solution through a plurality of iterations of the random initial value, and even if each iteration converges to a lower error value, the resulting solution contains negative values (there is no possibility that there is a negative value in the coverage). Therefore, in a complex catalytic system, the hybrid iterative algorithm has the advantages of being more accurate and stable in convergence compared with the traditional Newton method iteration.
Claims (6)
1. A heterogeneous catalysis surface coverage acquisition method combining ODE integration and Newton method iteration is characterized by comprising the following steps:
1) starting a micro-kinetic simulation of the heterogeneous catalysis;
2) randomly initializing a surface coverage and constructing a micro-dynamics steady-state equation;
3) taking the initialized coverage as a starting point, and integrating the steady-state equation by adopting an ODE numerical integration algorithm;
4) when the integral reaches the designated time, terminating ODE numerical integration, and obtaining a current solution as an initial value of Newton iteration to be sent into a Newton iterator for Newton iteration;
5) if the change rate of the coverage along with the time is smaller than the convergence precision, terminating the iteration, obtaining a steady state solution, performing the step 6), if the coverage is not converged, returning to the step 3), taking the current solution as a starting point of an ODE numerical integration algorithm, continuing the ODE numerical integration, and repeating the steps 3) -5);
6) and obtaining the reaction rate of the catalytic system according to the steady state solution.
2. The method for obtaining heterogeneous catalytic surface coverage according to claim 1, wherein in the step 2), the micro-dynamic steady state equation is as follows:
wherein the content of the first and second substances,
is adsorbate AnThe degree of coverage on the surface is,
is adsorbate AnCoefficient in elementary reaction e, reIs the reaction rate of the elementary reaction.
3. The method for obtaining the coverage of the heterogeneous catalytic surface by combining ODE integration and Newton method iteration as claimed in claim 1, wherein the convergence accuracy in the step 5) is 10E-50.
4. The method for obtaining heterogeneous catalytic surface coverage according to claim 1, wherein the step 4) is a step in which the time span and time step of ODE numerical integration are automatically adjusted according to the hybrid iteration, and comprises:
T＝α×10N
I＝10-(N+2)
wherein T is the time span of integration, I is the time step length, alpha is an adjustable value with the size of 0-1, and N is the number of times of current hybridization iteration.
5. The method for obtaining the coverage of the heterogeneous catalytic surface by combining ODE integration and Newton method iteration as claimed in claim 1, wherein in the step 3), the ODE numerical integration algorithm adopts a real-valued variable coefficient ODE integration solver.
6. The method for obtaining the heterogeneous catalytic surface coverage combining ODE integration and Newtonian iteration according to claim 1, wherein the step 4) comprises the following steps:
41) the solution of ODE numerical integration is taken as the initial coverage theta0；
42) Obtaining an iteration direction d:
43) the iteration rate constant lambda is equal to 0.5i-1As an iteration step length, wherein i is the step number of the current Newton iteration;
44) after obtaining the iteration direction and step length, pass Θi+1＝Θi+ λ d update coverage;
45) terminating the iteration when the iteration converges, otherwise returning to step 42) until convergence.
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